POLITECNICO DI MILANO Scuola di Ingegneria Industriale e …...Tanio, Lucas Yugo Converters for...

POLITECNICO DI MILANO

Scuola di Ingegneria Industriale e dell’Informazione Electrical engineering Master of Science Course

LUCAS YUGO TANIO

CONVERTERS FOR AUTOMOTIVE APPLICATIONS Multidevice Interleaved Boost Converter

Milano – MI 2018

LUCAS YUGO TANIO

CONVERTERS FOR AUTOMOTIVE APPLICATIONS

Multidevice Interleaved Boost Converter Thesis presented at Politecnico di Milano, as part of requirements to obtain the title of Master of Science in Electrical Engineering. Area: Power electronics, converters Supervisor: Prof. Roberto Perini

Milano – MI

2018

Tanio, Lucas Yugo

Converters for automotive applications / Lucas Yugo Tanio. – Milano, 2018.

102p.

Thesis Master of Science – Politecnico di Milano, Scuola di Ingegneria

Industriale e dell’Informazione

Supervisor: Prof. Roberto Perini

1. Hybrid electric vehicles. 2. DC/DC converters. 3. Small signal model; 4. Dual loop

control

Name: Lucas Yugo Tanio

Title: Converters for automotive applications

Thesis presented to the Master of Science program in Electrical Engineering at Scuola

di Ingegneria Industriale e dell’ Informazione of Politécnico di Milano

Approved thesis. Milano, _________________ ____, 2018.

EXAMINATION BOARD

________________________________________

Prof. Roberto Perini

Associated Professor at Politecnico di Milano

________________________________________

To my family and friends

who supported me in this journey.

Acknowledgements

I would like to express my gratitude to my thesis advisor, Professor Roberto

Perini, for the trust and assistance to develop this thesis. I would also like to thank the

Politecnico di Milano for the study opportunity and the support of my family and friends.

SOMMARIO

L'interesse per i veicoli ibridi elettrici è in crescita, non solo a causa dell'impatto

ambientale nel settore dei trasporti, ma anche perché sono economicamente

interessanti per quanto riguarda i costi di guida e manutenzione rispetto al motore a

combustione interna. L'utilizzo della pila a combustibile di tipo PEM come principale

fonte dei veicoli ibridi elettrici può rappresentare il futuro dell'industria automobilistica.

L’interfaccia tra la cella a combustibile e il motore elettrico è costituito da un

convertitore CC/CC e da un inverter trifase. In questo lavoro, viene presentato il

convertitore “Multidevice interleaved boost converter (MDIBC)” che è responsabile di

aumentare e regolare la tensione dal PEMFC al DC-link. Il convertitore proposto è

confrontato con altre due topologie di convertitori: il “Interleaved boost converter (IBC)”

e il “Multidevice boost converter (MDBC)”. Per le tre topologie, le equazioni del modello

di piccolo segnale e il design della strategia di controllo a doppio loop sono dimostrate

e illustrate utilizzando MATLAB e Simulink.

Parole chiavi: Veicoli elettrici ibridi; CC/CC convertitori; modello di piccolo segnale,

due loop controllo.

ABSTRACT

The interest in hybrid electric vehicles (HEV) is growing, not only because of

environmental impact in the transportation sector, but also because they are becoming

economically attractive regarding the cost of driving and maintenance when compared

to internal combustion engine. The use of PEM fuel cell (PEMFC) as a main source of

HEV can represent the future of the automotive industry. The link between the fuel cell

and the electric motor is made by a DC/DC converter and a three-phase inverter. In

this work, it is presented a unidirectional multidevice interleaved boost converter

(MDIBC) which is responsible to increase and regulate the voltage from the PEMFC to

the DC-link. The proposed converter is compared to other two converter topologies:

the interleaved boost converter (IBC) and the multidevice boost converter (MDBC). For

the three topologies, the equations of the small signal model and the design of dual

loop control strategy are demonstrated and illustrated using MATLAB and Simulink.

Key-words: Hybrid electric vehicles; DC/DC converters; small signal model; dual loop

control.

LIST OF FIGURES

Figure 1 – Schematic block diagram of a BEV powertrain ........................................ 20

Figure 2 – Schematic block diagram of a HEV powertrain ....................................... 20

Figure 3 – Ragone plot of different storage systems ................................................ 21

Figure 4 – Global greenhouse gas emissions by sector, based on global emissions

from 2010 .................................................................................................................. 26

Figure 5 – U.S. national average of the source of electricity and the annual

emissions per type of vehicle .................................................................................... 27

Figure 6 – California State average of the source of electricity and the annual


Figure 7 – Missouri State average of the source of electricity and the annual


Figure 8 – Generic electric block diagram of the fuel cell powertrain ........................ 30

Figure 9 – Electric circuit of the Multidevice Interleaved Boost Converter (MDIBC) . 30

Figure 10 – MDIBC gate sequence for the controllable switches when the duty cycle

𝑑 is 0.25..................................................................................................................... 33

Figure 11 – MDIBC Gate sequence for the controllable switches when the duty cycle

𝑑 is 0.10..................................................................................................................... 34


𝑑 is 0.40..................................................................................................................... 35


(d) is 0.50 .................................................................................................................. 36

Figure 14 – Position of poles and zeros of the transfer function 𝑖𝐿𝑠𝑑𝑠 ..................... 45

Figure 15 – Position of poles and zeros of the transfer function 𝑣𝑜𝑠𝑑𝑠 ..................... 45

Figure 16 – MDIBC Bode diagram transfer function of 𝑖𝐿𝑠𝑑𝑠 and 𝑣𝑜𝑠𝑑𝑠 .................. 46

Figure 17 – MDIBC Bode diagram of normalized transfer function 𝑖𝐿𝑠𝑑𝑠 ................. 46

Figure 18 – MDIBC Bode diagram of normalized transfer function 𝑣𝑜𝑠𝑑𝑠 ................ 47

Figure 19 – Absolute value of zeros and poles as function of the load resistance (𝑅)

.................................................................................................................................. 47

Figure 20 – Position of zeros and poles as function of the load resistance in the

complex plane. .......................................................................................................... 48

Figure 21 – Electric circuit of the Two-Phase of Interleaved Boost Converter (IBC) . 51

Figure 22 – IBC Gate sequence for the controllable switches when the duty cycle 𝑑

is 0.40 ........................................................................................................................ 51

Figure 23 – Position of poles and zeros of the transfer function 𝑖𝐿𝑠𝑑𝑠 ..................... 58

Figure 24 – Position of poles and zeros of the transfer function 𝑣𝑜𝑠𝑑𝑠 ..................... 59

Figure 25 – IBC Bode diagram of the inductor current and output voltage ............... 59

Figure 27 – IBC Bode diagram of output voltage normalized – unit gain .................. 60

Figure 28 – Electric circuit of the Multidevice boost converter (MDBC) .................... 61

Figure 29 – MDBC Gate sequence for the controllable switches when the duty cycle

𝑑 is 0.40..................................................................................................................... 61

Figure 30 – Position of poles and zeros of the transfer function 𝑖𝐿𝑠𝑑𝑠 for the MDBC

.................................................................................................................................. 67

Figure 31 – Position of poles and zeros of the transfer function 𝑣𝑜𝑠𝑑𝑠 for the MDBC

.................................................................................................................................. 67

Figure 32 – MDBC Bode diagram of the inductor current and output voltage........... 68

Figure 33 – MDBC Bode diagram of inductor current normalized – unit gain ........... 69

Figure 34 – MDBC Bode diagram of output voltage normalized – unit gain ............. 69

Figure 35 – Dual loop control strategy adopted. ....................................................... 71

Figure 36 – Inner loop control – current inductor control .......................................... 72

Figure 37 – Simplification of the inner loop control ................................................... 74

Figure 38 – 𝑘𝑝𝐼 𝑣𝑠. 𝜑𝑚𝐼 and 𝑘𝑖𝐼 𝑣𝑠. 𝜑𝑚𝐼 – MDIBC .................................................... 76

Figure 39 – 𝑘𝑝𝑉 𝑣𝑠. 𝜑𝑚𝑉 and 𝑘𝑖𝑉 𝑣𝑠. 𝜑𝑚𝑉 – MDIBC ................................................ 77

Figure 40 – 𝑘𝑝𝐼 𝑣𝑠. 𝜑𝑚𝐼 and 𝑘𝑖𝐼 𝑣𝑠. 𝜑𝑚𝐼 – IBC ......................................................... 78

Figure 41 – 𝑘𝑝𝑉 𝑣𝑠. 𝜑𝑚𝑉 and 𝑘𝑖𝑉 𝑣𝑠. 𝜑𝑚𝑉 – IBC ...................................................... 78

Figure 42 – 𝑘𝑝𝐼 𝑣𝑠. 𝜑𝑚𝐼 and 𝑘𝑖𝐼 𝑣𝑠. 𝜑𝑚𝐼 – MDBC ..................................................... 79

Figure 43 – 𝑘𝑝𝑉 𝑣𝑠. 𝜑𝑚𝑉 and 𝑘𝑖𝑉 𝑣𝑠. 𝜑𝑚𝑉 – MDBC ................................................. 80

Figure 44 – Transfer function GI – MDIBC ............................................................... 81

Figure 45 – Open loop LI(s) and closed loop KI(s) transfer functions – MDIBC ....... 81

Figure 46 – Transfer function GV – MDIBC .............................................................. 82

Figure 47 – Open loop LV(s) and closed loop KV(s) transfer functions – MDIBC ..... 82

Figure 48 – Transfer function GI – IBC ..................................................................... 83

Figure 49 – Open loop LI(s) and closed loop KI(s) transfer functions – IBC ............. 83

Figure 50 – Transfer function GV – IBC ................................................................... 84

Figure 51 – Open loop LV(s) and closed loop KV(s) transfer functions – IBC .......... 84

Figure 52 – Transfer function GI – MDBC ................................................................ 85

Figure 53 – Open loop LI(s) and closed loop KI(s) transfer functions – MDBC ........ 85

Figure 54 – Transfer function GV – MDBC ............................................................... 86

Figure 55 – Open loop LV(s) and closed loop KV(s) transfer functions – MDBC...... 86

Figure 56 – Electric circuit of the MDIBC in Simulink ................................................ 87

Figure 57 – Regulator block ...................................................................................... 87

Figure 58 – Start logic for the drive gate signal ........................................................ 88

Figure 59 – Controller diagram MDIBC in Simulink .................................................. 88

Figure 60 – Anti-windup PI controller ........................................................................ 89

Figure 61 – Electric circuit of the IBC in Simulink ..................................................... 90


Figure 63 – Electric circuit of the IBC in Simulink ..................................................... 91


Figure 65 – 𝑣𝑜 vs. t (orange curve) and 𝑣𝑜 𝑟𝑒𝑓 vs. t (blue curve) for the MDIBC ..... 91

Figure 66 – 𝑣𝑜vs. t (orange curve) and 𝑣𝑜 𝑟𝑒𝑓 vs. t (blue curve) for the MDIBC in

steady state condition for the MDIBC ........................................................................ 92

Figure 67 – 𝑖𝑑 𝑣𝑠. 𝑡 (green curve), 𝑖𝐿1𝑣𝑠. 𝑡 (blue curve) and 𝑖𝐿2𝑣𝑠. 𝑡 (orange curve) for

MDIBC ....................................................................................................................... 94

Figure 68 – 𝑖𝐿1𝑣𝑠. 𝑡 (blue curve) and 𝑖𝐿2𝑣𝑠. 𝑡 for MDIBC (orange curve) for MDIBC 94

Figure 69 – 𝑖𝑑 𝑣𝑠. 𝑡 for MDIBC .................................................................................. 94

Figure 70 – 𝑣𝑜vs. t (orange curve) and 𝑣𝑜 𝑟𝑒𝑓 vs. t (blue curve) for the IBC in steady

state condition ........................................................................................................... 95

Figure 71 – 𝑖𝑑 𝑣𝑠. 𝑡 (green curve), 𝑖𝐿1𝑣𝑠. 𝑡 (blue curve) and 𝑖𝐿2𝑣𝑠. 𝑡 (orange curve)

for IBC ....................................................................................................................... 95

Figure 72 – 𝑖𝐿1𝑣𝑠. 𝑡 (blue curve) and 𝑖𝐿2𝑣𝑠. 𝑡 for MDIBC (orange curve) for IBC ...... 96

Figure 73 – 𝑖𝑑 𝑣𝑠. 𝑡 for IBC ........................................................................................ 96

Figure 74 – 𝑣𝑜vs. t (blue curve) and 𝑣𝑜 𝑟𝑒𝑓 vs. t (orange curve) for the MDBC in

steady state condition ................................................................................................ 97

Figure 75 – 𝑖𝑑 𝑣𝑠. 𝑡 for MDBC ................................................................................... 97

LIST OF TABLES

Table 1 – Description of the MDIBC conduction of the IGBTs and diodes and load

situation when duty cycle 𝑑 is 0.25 during one period ............................................... 33







Table 5 – Steady state value of 𝐼𝐿 and 𝑉𝑜 for MDIBC .............................................. 43

Table 6 – Value of the converter components. .......................................................... 43

Table 7 – Position of the zero and poles of the state equations in the MDIBC ......... 44

Table 8 – Converter type identification through the value of number of phases 𝑛 and

the number of parallel legs per phase (𝑚) ................................................................. 50

Table 9 – Description of the IBC conduction of the IGBTs and diodes and load

situation ..................................................................................................................... 52

Table 10 – Steady state value of 𝐼𝐿 and 𝑉𝑜 for IBC .................................................. 57

Table 11 – Position of the zero and poles of the state equations in the IBC ............. 58

Table 12 – Description of the MDBC conduction of the IGBTs and diodes and load

situation ..................................................................................................................... 62

Table 13 – Steady state value of 𝐼𝐿 and 𝑉𝑜 for MDBC ............................................. 66

Table 14 – Position of the zero and poles of the state equations in the MDBC ........ 66

Table 15 – Efficiency and converter losses ............................................................... 70

Table 16 – Values of current and voltage cut-off frequency for each converter type. 75

Table 17 – Value of the controller proportional and integral parameters

𝑘𝑝𝐼, 𝑘𝑖𝐼 𝑘𝑝𝑉, 𝑘𝑖𝑉 for the MDIBC when 𝜑𝑚𝐼 = 60 𝑑𝑒𝑔, 𝜑𝑚𝑉 = 70 𝑑𝑒𝑔 ....................... 76


𝑘𝑝𝐼, 𝑘𝑖𝐼 𝑘𝑝𝑉, 𝑘𝑖𝑉 for the IBC when 𝜑𝑚𝐼 = 60 𝑑𝑒𝑔, 𝜑𝑚𝑉 = 60 𝑑𝑒𝑔 ............................. 77


𝑘𝑝𝐼, 𝑘𝑖𝐼 𝑘𝑝𝑉, 𝑘𝑖𝑉 for the MDBC when 𝜑𝑚𝐼 = 70 𝑑𝑒𝑔, 𝜑𝑚𝑉 = 70 𝑑𝑒𝑔 ........................ 79

Table 20 – Output voltage, total current and inductor current ripple for each topology

.................................................................................................................................. 98

CONTENTS

1. INTRODUCTION ................................................................................................ 18

1.1 Description of the multidevice interleaved boost converter .............................. 18

1.2 Chapters into which the subject is divided ....................................................... 19

1.3 History of electric vehicles ................................................................................ 19

1.4 Description of the state of art ........................................................................... 20

2. ELECTRIC VEHICLES AND THEIR ROLE IN THE SOCIETY ........................... 25

3. THE CONVERTER ............................................................................................. 30

3.1 Proposed converter .......................................................................................... 30

3.2 Switching power converter ............................................................................... 32

3.3 Small signal model ........................................................................................... 37

3.4 Current ripple in fuel cells and batteries ........................................................... 48

4. OTHER CONVERTERS ..................................................................................... 50

4.1 Two-phase interleaved boost converter (IBC) .................................................. 50

4.1.1 Switching power converter ......................................................................... 51

4.1.2 Small signal model ..................................................................................... 52

4.2 Multidevice boost converter (MDBC) ................................................................ 61

4.2.1 Switching power converter ......................................................................... 61

4.2.2 Small signal model ..................................................................................... 62

5. LOSSES ............................................................................................................. 70

6. SWITCHING POWER CONTROL ...................................................................... 71

6.1 Inner loop control ............................................................................................. 72

6.2 Outer loop control ............................................................................................. 74

7. SIMULATION AND RESULTS ............................................................................ 87

8. CONCLUSION ...................................................................................................... 99

9. REFERENCE ...................................................................................................... 100

18

1. INTRODUCTION

The electric vehicles are gaining popularity, not only because of environmental

impact of the transportation sector, such as climate change and urban pollution, but

also because they are convenient, quiet; it is for the advance of technology in batteries,

converters and electric motors they are becoming economically attractive since their

overall cost of maintenance and driving are cheaper when compared with the internal

combustion engine. Electric vehicles are reliable, safe and comfortable, turning it into

an interesting solution for our daily commuting. On the other hand, electric vehicle still

has to face with some problems as high purchase cost, long charging time – around

one hour – and lack of fuelling infrastructure which could represent an obstacle for long

range trip.

In some decades, however, concerning the public policies that are being

announced from different countries, internal combustion vehicles will no longer be able

to drive inside the city centre, therefore electric and hybrid vehicles will gradually

substitute them. The obsolescence of internal combustion vehicles will soon become

more evident.

1.1 Description of the multidevice interleaved boost converter

In this work, it is presented the design and control of unidirectional multidevice

interleaved boost converter that interfaces an energy storage system, and a DC link

(or a DC bus) that can be used in a battery electric vehicle (BEV) or a hybrid electric

vehicle (HEV) powertrain. This converter may have several applications due to its

performance and useful electric properties, such as low input current and output

voltage ripple and a necessity of decreased size and weight of the passive

components. Moreover, the converter controlling is not as cumbersome as other

practical solutions presented in this type of applications, e.g. Z – source inverter. In

this case, the converter was conceived to increase the voltage from the energy storage

system (e.g. fuel cell), connected in the converter input. Another role of the converter

would be the regulation of the output voltage, which means that the output voltage

should be a constant value, e.g. 400V, at any moment. This regulation must be

achieved by means of the proposed control.

19

For the sake of comparison, the multidevice interleaved boost converter will be

compared with similar other converters: two-phase interleaved boost converter (IBC)

and a multidevice boost converter (MDBC), in terms of the switching power converter

pattern, bode plot response, losses and value of passive components. These

comparisons will be made by means of calculations, plotting and simulations in

MATLAB and Simulink.

1.2 Chapters into which the subject is divided

This work is formed by seven main chapters organized as follows. In this chapter

it is presented an overview of what the work is about and the history of the electric car

and the trending technology of power converter for automotive applications. Then, in

the next chapter, it is discussed the role and the future of electric vehicles in the society.

In chapter three, it is deeply understood the proposed converter concerning the

circuitry, the drive gate signals and the small signal model is also presented. Moreover,

some details about energy storage system are also considered in this part. The study

of similar converters such as the two-phase interleaved boost converter (IBC) and the

multidevice boost converter (MDBC) is analysed in chapter four. In the next chapter, it

is investigated the losses in different converters presented in chapters three and four.

The switching power control is presented in chapter six. Finally, in chapter seven it is

validated the study by illustrating the simulation and results.

1.3 History of electric vehicles

According to the official website of U.S. Department of Energy [1], the history of

electric vehicles starts in the 18th century when researches from Hungary, Netherlands

and the United States of America proposed a concept of battery-powered vehicle.

However, the interest in this type technology starts in the 1970s with soaring oil prices

and gasoline shortages. In that time, the technology available for electric vehicles were

not competitive specially with respect to a limited performance when compared to

internal combustion cars. The range was limited to a few tens of kilometres before the

necessity of another recharge, while the top speed was limited to 70 kilometres per

hour. Then, only in the end of 20th century, with the development of technologies

around batteries and the microelectronic, a decisive moment came with the release of

the first mass produced hybrid electric car, the Toyota Prius in 1997 in Japan. However,

it became a success when it was released worldwide in the end of the year 2000. In

20

2006, another turning point, that brought a new concept around electric car, was the

announcement of a Silicon Valley start-up, Tesla motors, that would start producing

luxury electric sport-cars that could range, with only one recharge, 200 miles. Since

then, other electric car brands start to release new electric vehicles with different

technologies, such as, in 2010, the first commercially available plug-in hybrid electric

vehicles (PHEVs). Nonetheless, it is important to point that the spread of electric was

only possible in the countries whose government have invested in charging

infrastructure and contributed with tax initiatives.

1.4 Description of the state of art

The structure of the powertrain (i.e. the intervening mechanism by which power

is transmitted from an engine to a propeller or axle that it drives, [2]) of electric vehicles

is usually composed of four main elements: one or more electric storage systems which

have the function of a power source, a DC/DC converter that regulates the voltage of

a DC bus, a DC/AC inverter and the electric motor. The figures 1 and 2 show the block

diagram of a battery electric vehicle (BEV) and of hybrid electric vehicle (HEV)

respectively.

Figure 1 – Schematic block diagram of a BEV powertrain

Figure 2 – Schematic block diagram of a HEV powertrain

21

Note: In case the high energy storage system is a fuel cell, the double arrow, that represents the electric

flow becomes a right arrow, due to fuel cells electric unidirectional behaviour.

In the literature, some researches propose that powertrains should be fed by

different sources in order to exploit the strong features from them. Usually, these

sources present different characteristics: the main source is a high energy element,

which has the main function of feeding a common DC bus and consequently providing

enough energy to keep the electric motor working. On the other hand, the main aim of

secondary or auxiliary sources is assisting the main source during transient periods

and instantaneous peak power demand, such as acceleration. Hence, these auxiliary

sources should provide a high amount of energy during a short period, which means

that they must present as main characteristic a high-power density. The combination

of these two different storage systems not only make available energy and power

capabilities, therefore achieving a desired performance, but also this hybridization

helps to improve the life span of the main source [3], [4].

There are many arrangements possible to integrate different storage systems

based on their characteristics, the relation of power and energy density of different

storage systems can be seen in the “Ragone Plot”, figure 3.

Figure 3 – Ragone plot of different storage systems

22

Source: J. W. Shim, Y. Cho, S. Kim, S. W. Min and K. Hur, "Synergistic Control of SMES and Battery

Energy Storage for Enabling Dispatchability of Renewable Energy Sources," in IEEE Transactions on

Applied Superconductivity, vol. 23, no. 3, pp. 5701205-5701205, June 2013, Art no. 5701205. [6]

From the figure 3, it is possible that, for example, the Li-ion Batteries are a type

of storage system with a high energy density and average power density, when

compared with the others storage systems. Hence, a possible configuration as a

source for the powertrain would be the use of only this type of storage system, which

would configure a battery electric vehicle (BEV) as proposed in [5]. Even if the use of

only one type of storage system could be a practical solution, it would imply a large

battery pack, which means an increase of volume, weight and cost. These increases

could represent constraints in order to make the electric vehicle more efficient and

market acceptable. On the other hand, the use of two complementary different sources

can overcome the power deficiency at a lower volume, weight and cost. Some authors

propose a hybridization that consists in batteries as main sources and ultracapacitors

([3] - [7]) or supercapacitors [4] as auxiliary source. Other authors have also proposed

a fuel cell hybrid electric vehicle [8]: in this case, the fuel cell would be the main source

and another energy storage system, such as a Li-ion battery, would aid the first source.

According to [4], to downsize the storage system, e.g. the battery pack, a low

voltage (LV) source is connected to a DC bus through a DC/DC converter. As the DC

bus voltage characteristic is higher than the output voltage of the storage system, the

main function of the DC/DC converter is to increase and regulate the voltage in the DC

bus. The main improvement of this choice is, in case of a bidirectional converter, the

ability to recover more energy from the DC-link during the regenerative braking than

HV source that is directly connected to the DC bus.

The converter that interfaces the DC bus and the energy storage system is a

keystone in order to combine the properties of two different sources.

There are several ways to do the DC/DC converter, the simplest one is the use

of boost converter. In [7], the authors have proposed a similar solution, in which parallel

buck-boost converters are used: the boost configuration is used when the vehicle is in

traction mode, otherwise the buck mode is used during regenerative braking.

Moreover, the input of each converter is different, in this case ultracapacitor and

battery. It is also possible to use variations of the boost converter, such as two-phase

23

interleaved boost converter (IBC), as proposed in [9] concerning a ripple cancellation

of the two different phases. Other possible configuration that is also mentioned in the

literature is the multidevice boost converter (MDBC) which by analogy is compared to

IBC. The multidevice interleaved boost converter (MDIBC) is a configuration that

combines the properties of IBC and MDBC configurations; thus it can enjoy

advantages from both configurations, for example reducing the sizes and weight of

passive components, low input current and low output voltage ripples and high-power

factor.

Regarding the DC/AC inverter there are two interesting solutions that could

represent the future in electric vehicles. The first one is a Z-source inverter, as

proposed in [10]. This solution would have the role not only as an inverter, but also as

a buck-boost converter, which means that the Z-source inverter would interface the

battery and the motor without the need of the two stages of conversion. The necessity

of only one stage of conversion implies a lower number of components, therefore a

lower size and cost when compared to the two-stage solution. However, as mentioned

in [5], the drawbacks in this configuration must be considered in the design circuit

process, the most cumbersome are the high current and voltage stress, the necessity

of a complex control and a limit boost ratio. The second solution is proposed in [11]

and used in [5] and [8]; it is a bidirectional DC/AC converter made of eight switches,

by this reason the solution name is eight switches inverter. This converter can work in

four different modes: it can work as a three-phase AC/DC inverter when the motor is

working in traction mode, as a three-phase DC/AC rectifier in case of regenerative

braking and two other single-phase modes that relates the onboard battery and the

power from the grid.

The last element of the block diagram of electric vehicle, as shown in the figures

1 and 2, is the electric motor. In spite of controlling AC motors precisely, the articles

[5] and [11] proposed two possible strategies, the first, and also the most popular, is

the indirect field oriented control (IFOC) in which the flux and torque currents are

decoupled and controlled through vector control; however, for a constant rotor flux, it

features poor efficiency of the motor at low load. Hence, in the interest of optimizing

the efficiency of the induction motor, they have proposed an IFOC based on particle

24

swarm optimization (PSO). The PSO, in this case, is used to assess the optimal flux

that minimizes the total losses.

25

2. ELECTRIC VEHICLES AND THEIR ROLE IN THE SOCIETY

The concerns about climate change have become an issue to be solved for the

world community due to not only significant impact in the earth’s climate but also due

to severe consequences in the public health and economic implications. In 1979, the

humanity took the first step, as an international concern, to combat the climate change

through the First World Climate Conference. In this conference the World Climate

Program was established which gave the guidelines in climate research, monitoring,

applications and impact assessment, as written in the “World Meteorological

Organization” website [12]. The same program was reviewed and refocused in the

Second World Climate Conference in 1990. Since then, a lot of effort was put from

several countries in order to propose agreements and milestones for the future years:

as reference, it is possible to cite the Kyoto Protocol in 1997 and the Paris Agreement

in 2015. The Paris Agreement, for example, was able to join 195 nations into the mutual

cause of combating climate change through investments and public policies towards a

sustainable future. According to United Nations Climate Change webpage [13], “The

Paris Agreement central aim is to strengthen the global response to the threat of

climate change by keeping a global temperature rise this century well below 2 degrees

Celsius above pre-industrial levels and to pursue efforts to limit the temperature

increase even further to 1.5 degrees Celsius. Additionally, the agreement aims to

strengthen the ability of countries to deal with the impacts of climate change. To reach

these ambitious goals, appropriate financial flows, a new technology framework and

an enhanced capacity building framework will be put in place, thus supporting action

by developing countries and the most vulnerable countries, in line with their own

national objectives. The Agreement also provides for enhanced transparency of action

and support through a more robust transparency framework.”

In the same direction, some countries have announced severe public policies

regarding internal combustion vehicles. In France, by 2040 petrol and diesel vehicles

no longer will on be sale. Norway, country in which around 40% of the sales in 2016

were electric or hybrid, is planning that in 2025 all new passenger vehicle must be

zero-emissions. Other European countries such as Germany, Spain and Netherlands,

or non-European such as China and India are planning aggressive polices aiming at

the hybrid and electric vehicles as a solution of the gases prevenient from the

transportation sector.

26

The transportation sector is responsible for 14% of the global greenhouse gas

emissions, according to the data Climate Change 2014: Mitigation of Climate Change,

as shown in the figure 4.

Figure 4 – Global greenhouse gas emissions by sector, based on global emissions

from 2010

Source: “Global Greenhouse Gas Emissions Data”. United States Environmental Protection Agency.

Available in: https://www.epa.gov/. Access in: November 16th, 2018. [14]

There are two different classes of vehicle emissions, as mentioned in [15]. The

first category is the direct emission that come directly from the evaporation from the

fuel system and during the fuelling process. It also includes smog-forming pollutants,

other pollutants harmful to human health and greenhouse gases. The second category

is the life cycle emissions, that is related to fuel and vehicle production, processing,

distribution, use and recycling or disposal. It is important to notice that electric vehicles

do not cause direct emissions. Moreover, as the life cycle emission is variable since it

depends on the energetic matrix in which the vehicle and the fuel is produced, the

produced emission of any type of vehicle also varies as the production location

changes. The calculation of the life cycle emission is quite cumbersome, due to the

several variables to be considered. However, in the U.S. Department of Energy –

Energy Efficiency & Renewable Energy website, it is possible to visualize the annual

emission per vehicle as a national average or concerning each state, as the electricity

27

sources change, therefore a deeply analysis becomes easier to be made. Examples

regarding the national, the California State and the Missouri State annual emissions

average per vehicle are shown in the figures 5, 6 and 7 respectively.

Comparing the figures 6 and 7, it is possible to understand that all electric

vehicles are worth it, considering only the annual emissions per vehicle. In the

California State, where 47.53% of the produced energy comes from renewable sources

(hydro, solar, wind, geothermal and Biomass), all electric vehicles have a lower annual

emission when compared to the other types of vehicle. On the other hand, in the state

of Missouri, where 79.80% of the produced electricity comes from coal, a non-

renewable energy, all electric and plug-in hybrid vehicles present an annual emission

per vehicle higher than hybrid vehicles. All electric vehicles in the State of Missouri can

emit around three times more than the State of California. Hence, as shown in the

figure 5, as a national average the order of the least to the higher emitter are all electric,

plug-in hybrid and hybrid vehicles, the same sequence happen in the California State,

but it is different for the Missouri State, where hybrid vehicles are less emitter than the

all-electric and plug in hybrid.

Figure 5 – U.S. national average of the source of electricity and the annual emissions

per type of vehicle

Source: “Alternative Fuels Data Center - Emissions from Hybrid and Plug-In Electric Vehicles”. U.S.

Department of Energy. Available in: https://www.energy.gov/. Access in: November 16th, 2018. [16]

https://www.energy.gov/

28

Figure 6 – California State average of the source of electricity and the annual

emissions per type of vehicle



Figure 7 – Missouri State average of the source of electricity and the annual emissions

per type of vehicle





29

In a study [17] supported by New England University Transportation Center at

Massachusetts Institute of Technology (MIT), named “Can today’s EVs make a dent in

climate change?”, have reached a conclusion that electric vehicles can make the

difference for a sustainable future. According to the study, even if the car must be

recharged at least once a day (e.g. probably overnight) in the United States of America,

around 90% of personal vehicles necessity in the United States could be met by a “low-

cost electric vehicle on the market today”, considering the overall cost of the vehicle

and distance range. That is because technology regarding the distance range of the

electric vehicle and batteries not only are becoming cheaper, but also it has already

overcome the travelled distance of daily commute. This replacement would represent

a reduction of 30% in the emissions of the transportation sector and would meet the

climate objective targets for personal vehicle travel considering the current energetic

matrix of the country. If the United States energetic sources turn into the direction of

cleaner and renewable energy, the reduction percentual has the tendency to increase.

30

3. THE CONVERTER

The proposed converter studied in this paper is the unidirectional Multidevice

Interleaved Boost Converter (MDIBC), based in [8]. It is a DC/DC converter used to

increase the voltage that come from the energy storage system. In this work it is

presented a hybrid electric powertrain, where the main source is a Fuel Cell and the

auxiliary source is a battery. As the converter is a unidirectional converter, it cannot be

used to recover energy from the regenerative braking, then it is suitable for energy

storage systems that cannot absorb electric energy from their output terminals.

Figure 8 – Generic electric block diagram of the fuel cell powertrain

It is assumed that the load current is ripple free, the proposed converter

operates in the continuous conduction mode (CCM) and the switches have identical

duty cycles (𝑑1 = 𝑑2 = 𝑑3 = 𝑑4).

3.1 Proposed converter

The electric circuit of the unidirectional multidevice interleaved boost converter

(MDIBC) is presented in the figure 9.

Figure 9 – Electric circuit of the Multidevice Interleaved Boost Converter (MDIBC)

31

The converter is composed of one voltage source, two phases, four legs of

diodes in series of IBGTs, a shunt capacitor and the load. Each phase is connected to

the same source – through an inductor. From each phase there are two legs of diodes

in series with a controllable switch. For example, the first phase (𝑟𝐿1 and 𝐿1) is

connected to 𝐷1, 𝐺1, 𝐷2, 𝐺2. The second phase (𝑟𝐿2 and 𝐿2) is connected to the

remaining components (to 𝐷3, 𝐺3, 𝐷4, 𝐺4). It is important to notice that the load,

represented by a resistance (𝑅), is the DC bus which is connected to a DC/AC inverter

that supplies the electric motor (not represented in figure 8). In case of a HEV,

connected to the DC link would be the output of another MDIBC.

The circuit presented in the figure 9 can only conduct current from the source to

the load, therefore the source cannot absorb energy from the load in order to recharge

the source, situation that would happen in case of regenerative braking.

The use of a DC/DC converter is related to the size reduction of the storage

system. Theoretically, a high voltage source could be directly connected to the DC/AC

inverter, however, it would demand a battery of considerable size which would imply

an additional cost. A practical solution (cheaper and lighter) is the use of a low voltage

source connected to a DC/DC converter to increase and regulate the voltage value of

the DC bus.

The proposed converter, MDIBC, besides the step-up property, for the same

switching frequency, increases the inductor current and the output voltage ripple

frequency. This frequency increment contributes to a higher system bandwidth and

consequently a faster dynamic response for the converter [8]. In order to obtain an

increased frequency ripple and a phase-shift interleaved control the gate signal

sequence must be chosen properly. Considering that 𝑚 is the number of parallel power

switches per channel and 𝑛 is the number of channels or phases, the gate sequence

for the controllable switches is shifted by 360𝑜

𝑛𝑚. The input current ripple and output

voltage ripple 𝑚𝑛 times of the switching frequency. Therefore, according to [8], a

reduction of 𝑚 times in the size of the passive components is expected when the

proposed converter is compared with the 𝑛-phase interleaved DC/DC converters. The

reduction of passive components is a key element for the reduction of size, cost and

weight of the power converter, as the passive components – specially the power

32

inductor – are the heaviest components, hence it is preferable a small inductance value

in the interest of achieving the size and weight wanted.

Finally, another converter property is the following: if phase-shift interleaved

control is used, the sum of the current ripple in both phases results in ripple

cancellation, which means that the ripple in energy storage system will be lower than

expected when this solution is compared to a non-phase-shift interleaved control.

3.2 Switching power converter

As mentioned previously, the drive gate signal sequence is important to achieve

a phase-shift interleaved control. In the figure 10, it is shown the gate signal pattern

when the duty cycle (𝑑) is equal to 0.25. When the duty cycle assumes this value, as

there are four controllable switches, at any moment during one period, there is at least

an IGBT conducting. Table 1 describes which IGBT and diodes are conducting during

one period and the load situation - powered or not - in each time interval. It is important

to notice that the reference name for the diodes are shown in the figure 9.

Next graphs (figures 11, 12 and 13) show the drive gate signals in three different

cases, when the duty cycle is equal to 0.1, 0.4 and 0.5. The description of which

switches (diodes and IGBTs) are conducting and the load situation is shown in the

tables 2, 3 and 4 respectively.

In the first case (𝑑 = 0.1), as the duty cycle is lower than 0.25, there are only

two possible situations: no switch is powered or only one switch is conducting.

Therefore, the load at any interval is being powered. In the second case (𝑑 = 0.4),

when the duty cycle is greater than 0.25, there are also two different situations: only

one switching is conducting or an overlapping of gate signals from different phases. In

the first situation, the load is fed as in the previous cases, but when there are two

switches from different phases conducting in the same time, the current does not feed

the load. The third case, when the duty cycle is (𝑑 = 0.5), at any time interval there is

an overlapping between switches from different phases, then, the load cannot be fed.

This last case is the boundary value that duty cycle can assume. Hence the duty cycle

interval is 0 < 𝑑 < 0.5.

33

Figure 10 – MDIBC gate sequence for the controllable switches when the duty cycle

(𝑑) is 0.25


situation when duty cycle (𝑑) is 0.25 during one period

Time instant IGBT Diode Load

0 < 𝑡 𝑇𝑠⁄ < 0.25 𝐺1 is conducting 𝐷3, 𝐷4 are conducting Powered

0.25 < 𝑡𝑇𝑠⁄ < 0.50 𝐺3 is conducting 𝐷1, 𝐷2 are conducting Powered

0.50 < 𝑡𝑇𝑠⁄ < 0.75 𝐺2is conducting 𝐷3, 𝐷4 are conducting Powered

0.75 < 𝑡𝑇𝑠⁄ < 1 𝐺4 is conducting 𝐷1, 𝐷2 are conducting Powered

34


(𝑑) is 0.10




0 < 𝑡 𝑇𝑠⁄ < 0.10 𝐺1 is conducting 𝐷3, 𝐷4 are conducting Powered

0.10 < 𝑡𝑇𝑠⁄ < 0.25 None 𝐷1, 𝐷2, 𝐷3, 𝐷4 are conducting Powered




35



0.85 < 𝑡𝑇𝑠⁄ < 1 None 𝐷1, 𝐷2, 𝐷3, 𝐷4 are conducting Powered


(𝑑) is 0.40




0 < 𝑡 𝑇𝑠⁄ < 0.15 𝐺4, 𝐺1 are conducting None Not powered

36


0.25 < 𝑡𝑇𝑠⁄ < 0.40 𝐺1, 𝐺3 are conducting None Not powered





0.90 < 𝑡𝑇𝑠⁄ < 1 𝐺4 is conducting 𝐷1, 𝐷2 are conducting Powered


(d) is 0.50

37



Time instant IGBT Diode Load situation

0 < 𝑡 𝑇𝑠⁄ < 0.25 𝐺4, 𝐺1 are conducting None Not powered



0.75 < 𝑡𝑇𝑠⁄ < 1 𝐺2, 𝐺4 are conducting None Not powered

3.3 Small signal model

The small signal model was found using the linearization of the power stage

using state-space averaging as proposed in [18]. Considering the figures 9 and 11 as

the electric circuit and gate signal pattern reference, a null value of capacitance

resistance – the output voltage (𝑣𝑜) is equal to the voltage capacitor (𝑣𝑐) – it is possible

to derive the small signal model. As shown in the table 2, during one period, there are

eight possible states that the circuit can assume. However, as there are two switches

per phase, the states started after the time interval 𝑇𝑠

𝑚 are a repetition of the previous

states. Moreover, the duty cycle of every controllable switch in the circuit is assumed

identical, hence there are four possible states that the MDIBC can assume.

As can be seen in the figure 9, there are three energy storage passive elements

in the circuit – two inductors and one capacitor – therefore, three equations are

necessary to describe the circuit: one for each current inductor and another for the

voltage capacitor.

The four possible states are described by means of the Kirchhoff voltage law

(KVL) and Kirchhoff current law (KCL). Then, the system of equations is written in

matrix form and the derivative of inductor current and voltage capacitor are isolated in

the left side, i.e., in the same form as: 𝑝𝑥 = 𝐴𝑥 + 𝐵𝑣𝑑, where 𝑝 is the derivative

38

operator, 𝐴 is a three by three matrix, 𝐵 is a three by one matrix, 𝑡1 and 𝑡2are,

respectively, the time that the 𝐺1 and 𝐺3 go to zero.

• Case 1, 0 < t < t1, 𝑟𝑐 ≈ 0, 𝐺1 is on:

𝑉𝑑 − 𝐿. 𝑝𝑖𝐿1 − 𝑟𝐿 . 𝑖𝐿1 = 0𝑉𝑑 − 𝐿. 𝑝𝑖𝐿2 − 𝑟𝐿 . 𝑖𝐿2 − 𝑣𝑜 = 0

𝑖𝐿2 = 𝐶. 𝑝𝑣𝑜 +𝑣𝑜𝑅

(1)

𝑝𝑥1 = 𝐴1𝑥1 + 𝐵1𝑉𝑑

𝑝(𝑖𝐿1𝑖𝐿2𝑣𝑜

) =

(

−𝑟𝐿𝐿

0 0

0 −𝑟𝐿𝐿

−1

𝐿

01

𝐶−1

𝑅𝐶)

(𝑖𝐿1𝑖𝐿2𝑣𝑜

) +

(

1

𝐿1

𝐿0)

𝑉𝑑

• Case 2, t1 < t <T𝑠

4, 𝑟𝑐 ≈ 0:

𝑉𝑑 − 𝐿. 𝑝𝑖𝐿1 − 𝑟𝐿 . 𝑖𝐿1 − 𝑣𝑜 = 0𝑉𝑑 − 𝐿. 𝑝𝑖𝐿2 − 𝑟𝐿 . 𝑖𝐿2 − 𝑣𝑜 = 0

𝑖𝐿1 + 𝑖𝐿2 = 𝐶. 𝑝𝑣𝑜 +𝑣𝑜𝑅

(2)


𝑝(𝑖𝐿1𝑖𝐿2𝑣𝑜

) =

(

−𝑟𝐿𝐿

0 −1

𝐿

0 −𝑟𝐿𝐿

−1

𝐿1

𝐶

1

𝐶−1

𝑅𝐶)


) +

(

1

𝐿1

𝐿0)

𝑉𝑑

• Case 3, T𝑠

4< t < t2, 𝑟𝑐 ≈ 0, 𝐺3 is on:

𝑉𝑑 − 𝐿. 𝑝𝑖𝐿1 − 𝑟𝐿 . 𝑖𝐿1 − 𝑣𝑜 = 0𝑉𝑑 − 𝐿. 𝑝𝑖𝐿2 − 𝑟𝐿 . 𝑖𝐿2 = 0

𝑖𝐿1 = 𝐶. 𝑝𝑣𝑜 +𝑣𝑜𝑅

39


𝑝 (𝑖𝐿1𝑖𝐿2𝑣𝑜

) =

(

−𝑟𝐿𝐿

0 −1

𝐿

0 −𝑟𝐿𝐿

0

1

𝐶0 −

1

𝑅𝐶)


) +

(

1

𝐿1

𝐿0)

𝑉𝑑

(3)

• Case 4, 𝑡2 < t <T𝑠

2, 𝑟𝑐 ≈ 0:


𝑖𝐿1 + 𝑖𝐿2 = 𝐶𝑝𝑣𝑜 +𝑣𝑜𝑅

(4)

𝑝



) =

(

−𝑟𝐿𝐿

0 −1

𝐿

0 −𝑟𝐿𝐿

−1

𝐿1

𝐶

1

𝐶−1

𝑅𝐶)


) +

(

1

𝐿1

𝐿0)

𝑉𝑑

Described the state matrices, it is necessary to determine the average value of

derivative (𝑝 < 𝑥 >) of the state variables. An index will be written in order to identify

which case the matrices are from: for example, matrices A and B of the first case, will

receive an index 1. As there are a repetition after T𝑠

𝑚, the integration interval is 0 < t <

T𝑠

𝑚.

p < x >=1

T𝑠𝑚

∫ 𝑝𝑥. 𝑑𝑡

T𝑠𝑚

0

(5)

Simplifying the notation, i.e., the 𝑥 = < 𝑥 >, and substituting the values of px:

px =𝑚

T𝑠(∫ 𝑝𝑥1. 𝑑𝑡

𝑡1

0

+∫ 𝑝𝑥2. 𝑑𝑡

T𝑠4

𝑡1

+∫ 𝑝𝑥3. 𝑑𝑡𝑡2

T𝑠4


T𝑠𝑚

𝑡2

)

(6)

The second switch starts to conduct at 𝑇𝑠

𝑚𝑛, the number four in the denominator

is related to the number of switches in the circuit. Moreover, the number of switches is

40

related to the number of phases 𝑛 and to the number of parallel legs per phase 𝑚.

Then, the integration interval is rewritten.

px =𝑚

𝑇𝑠(∫ (𝐴1𝑥 + 𝐵1𝑣𝑑). 𝑑𝑡

𝑑.𝑇𝑠

0

+∫ (𝐴2𝑥 + 𝐵2𝑣𝑑). 𝑑𝑡

T𝑠𝑚𝑛

𝑑𝑇𝑠

+∫ (𝐴3𝑥 + 𝐵3𝑣𝑑). 𝑑𝑡𝑇𝑠(

1𝑚𝑛

+𝑑)

T𝑠𝑚𝑛

+∫ (𝐴4𝑥 + 𝐵4𝑣𝑑). 𝑑𝑡

T𝑠𝑚

𝑇𝑠(1𝑚𝑛

+𝑑)

)

px =𝑚

T𝑠((𝐴1𝑥 + 𝐵1𝑣𝑑). 𝑑T𝑠 + (𝐴2𝑥 + 𝐵2𝑣𝑑). (

1

𝑚𝑛− 𝑑) T𝑠 + (𝐴3𝑥 + 𝐵3𝑣𝑑). 𝑑T𝑠

+ (𝐴4𝑥 + 𝐵4𝑣𝑑). (1

𝑚𝑛− 𝑑)T𝑠)

px = 𝑚(((𝐴1 + 𝐴3)𝑑 +1

𝑚𝑛(𝐴2 + 𝐴4)(1 − 𝑚𝑛𝑑))𝑥

+ ((𝐵1 + 𝐵3)𝑑 +1

𝑚𝑛(𝐵2 + 𝐵4)(1 − 𝑚𝑛𝑑)) 𝑣𝑑)

(7)

Introducing the separation of AC perturbation and DC component in the

variables 𝑥, 𝑑. The letter tilde on top represents the AC perturbation and the capital

letter the DC component. The input voltage is considered pure DC.

𝑥 = + 𝑋𝑑 = + 𝐷𝑣𝑑 = 𝑉𝑑

(8)

Substituting the equations in (8) in (7), after algebraic manipulations and

neglecting higher order harmonics (e.g.: . ≈ 0).

𝑝 = 𝑚 [((𝐴1 + 𝐴3)𝐷 +1

𝑚𝑛(𝐴2 + 𝐴4)(1 − 𝑚𝑛𝐷))𝑋 + ((𝐵1 + 𝐵3)𝐷 +

1

𝑚𝑛(𝐵2 + 𝐵4)(1 − 𝑚𝑛𝐷))𝑉𝑑] + [((𝐴1 + 𝐴3)𝐷 +

1

𝑚𝑛(𝐴2 + 𝐴4)(1 − 𝑚𝑛𝐷)) +

((𝐴1 + 𝐴3 − 𝐴2 − 𝐴4)𝑋 + (𝐵1 + 𝐵3 − 𝐵2 − 𝐵4)𝑉𝑑)]

(9)

The first bracket in the equation 9, is the steady state response of the circuit

and, by definition, its value is zero. The matrices 𝐵1, 𝐵2, 𝐵3 and 𝐵4 are equal. Matrices

41

𝐴2 and 𝐴4 are also equal, because they represent the state when there is no switch

on, the sum of them is two times the value of 𝐴2. In this case, as there are two identical

matrices due to the fact that there are two-phases, it is also possible to substitute the

value “2” by “n”. Considering these prepositions, (9) can be rewritten.

𝑝 = 𝑚𝐴 + 𝑚(𝐴1 + 𝐴3 − 𝑛𝐴2)𝑋

𝐴 = (𝐴1 + 𝐴3)𝐷 +1

𝑚𝐴2(1 − 𝑚𝑛𝐷)

(10)

Finally, performing the Laplace transform and isolating the AC perturbations in

the left side, it is possible to find the transfer function of the state variables. The matrix

“I” is the unit matrix and “s” is the Laplace variable.

(𝑠)

(𝑠)= (𝑠𝐼 − 𝑚𝐴)−1𝑚(𝐴1 + 𝐴3 − 𝑛𝐴2)𝑋

𝐴 = (𝐴1 + 𝐴3)𝐷 +1

𝑚𝐴2(1 − 𝑚𝑛𝐷)

(11)

Substituting the values, it is possible to find the matrices values.

𝐴 =

(

−𝑟𝐿2𝐿

0 −1

2𝐿(1 − 2𝐷)

0 −𝑟𝐿2𝐿

−1

2𝐿(1 − 2𝐷)

1

2𝐶(1 − 2𝐷)

1

2𝐶(1 − 2𝐷) −

1

2𝑅𝐶 )

𝐴1 + 𝐴3 − 𝑛𝐴2 =

(

0 0

1

𝐿

0 01

𝐿

−1

𝐶−1

𝐶0)

In the matrix 𝑋 = (𝑥1 𝑥2 𝑥3)𝑡, there are the steady state variables of the

current inductors and the output voltage (𝑥1 = 𝐼𝐿1, 𝑥2 = 𝐼𝐿2, 𝑥3 = 𝑉𝑜) . This means that

the transfer functions 𝐿(𝑠)

(𝑠) and

𝑜(𝑠)

(𝑠) solutions are going to be around the values in the

matrix 𝑋. The current inductor steady state value (equation 12) is calculated

considering that the output power (𝑃𝑜) is equal to the input power (𝑃𝑖) and the current

inductor in both phases are the same. The steady state output voltage is calculated

according to the equation 13, in which non-ideal characteristics are neglected.

42

𝑃𝑖 = 𝑃𝑜

𝑃𝑖 = 𝑉𝑑𝐼𝐿1 + 𝑉𝑑𝐼𝐿2𝐼𝐿1 = 𝐼𝐿2 = 𝐼𝐿

𝐼𝐿 = 𝑃𝑜2𝑉𝑑

(12)

𝑉𝑜 = 𝑉𝑑

1 −𝑚𝐷

(13)

Performing some algebraic calculations, it is found the small signal perturbation

in the current inductor and the voltage capacitor.

• Current inductor transfer function:

𝐺𝑖𝑑(𝑠) =𝑖𝐿(𝑠)

(𝑠)= 𝐺𝑑𝑖

(1 +𝑠𝜔𝑧𝑖

)

∆(𝑠)

(14)

𝐺𝑑𝑖 =

2(𝑥3 + (𝑥1 + 𝑥2)(1 − 2𝐷)𝑅)

𝑟𝐿 + 2𝑅(1 − 4𝐷(1 − 𝐷))

𝜔𝑧𝑖 =𝑥3 + (𝑥1+𝑥2)(1 − 2𝐷)𝑅

𝑅𝐶𝑥3

∆(𝑠) =𝑠2

𝜔02 +

𝑠

𝑄𝜔0+ 1

𝜔0 = √𝑟𝐿 + 2𝑅(1 − 4𝐷(1 − 𝐷))

𝑅𝐿𝐶, 𝑄 =

√𝑅𝐿𝐶 (𝑟𝐿 + 2𝑅(1 − 4𝐷(1 − 𝐷)))

𝑅𝐶𝑟𝐿 + 𝐿

• Output voltage state equation:

𝐺𝑣𝑑(𝑠) =𝑜(𝑠)

(𝑠)= 𝐺𝑑𝑣

(1 −𝑠𝜔𝑧𝑣

)

∆(𝑠)

(15)

𝐺𝑑𝑣 =

2𝑅((1 − 2𝐷)𝑥3 − (𝑥1 + 𝑥2)𝑟𝐿)

𝑟𝐿 + 2𝑅(1 − 4𝐷(1 − 𝐷))

𝜔𝑧𝑣 =2(1 − 2𝐷)𝑥3 − 𝑟𝐿(𝑥1 + 𝑥2)

𝐿(𝑥1 + 𝑥2)

The value of the input voltage (𝑉𝑑) is set to 200V and the output power (𝑃𝑜) to

30 KW. Therefore, considering the duty cycle (D) is 0.25, from the equations 12 and

43

13, it is possible to calculate the steady state values of 𝐼𝐿 and 𝑉𝑜, these values are

shown in the table 5.

Table 5 – Steady state value of 𝐼𝐿 and 𝑉𝑜 for MDIBC

Parameter Value

𝐼𝐿 (A) 75

𝑉𝑜 (V) 400

Two parameters were adopted to choose the components values, the maximum

current inductor ripple allowed limited to 9% (∆𝐼𝐿

𝐼𝐿< 9%), and the maximum output

voltage ripple limited to 1% (∆𝑉𝑜

𝑉𝑜< 1%). The equations (16) to (19) show the equations

that calculate the output resistance (𝑅), the inductor (𝐿), the capacitor (𝐶) and the

inductor resistance (𝑟𝐿). It is important to point out that the value of the inductor

resistance was calculated considering a quality factor of Q|𝑓𝑠=1500 𝐻𝑧 = 100. The

component values are described in the table 6.

𝑅 =𝑉𝑜2

𝑃

(16)

𝐿 =𝑉𝑖𝑛𝐷

𝑓𝑠∆𝐼𝐿

(17)

𝐶 =𝐷

𝑓𝑠𝑅

𝑉𝑜∆𝑉𝑜

(18)

𝑟𝐿 =𝜔𝐿

𝑄

(19)

Table 6 – Value of the converter components.

Component Value

L 375 µH/180A

44

𝑟𝐿 34 mΩ

C 320 µF/500V

R 5.33 Ω

Therefore, in order to understand the transfer function shape and properties, the

position of zeros and poles is calculated and described in the table 7. The roots of the

numerator and the denominator gives the position of zero and poles respectively. The

poles position is equal for both state variables, as their denominator transfer function

are the same. Figures 14 and 15 illustrate the position of zero and poles in the complex

plane, for the transfer functions 𝐿(𝑠)

(𝑠) and

𝑜(𝑠)

(𝑠).

Table 7 – Position of the zero and poles of the state equations in the MDIBC

Parameter Value

𝑍𝑖 −1172 𝑟𝑎𝑑 𝑠⁄

𝑍𝑣 −6930 𝑟𝑎𝑑 𝑠⁄

𝑃1, 𝑃2 −338 ± 2026𝑗 𝑟𝑎𝑑 𝑠⁄

|𝑃1| = |𝑃2 | 2054 𝑟𝑎𝑑 𝑠⁄

𝜁 0.1668

Note: Subscript description: 𝑍 – zero, 𝑃 – pole, 𝑖 – inductor current, 𝑣 – output voltage,

𝜁 - damping ratio.

45

Figure 14 – Position of poles and zeros of the transfer function 𝐿(𝑠)

(𝑠)

Figure 15 – Position of poles and zeros of the transfer function 𝑜(𝑠)

(𝑠)

46

Figure 16 shows the Bode diagram of the 𝐿(𝑠)

(𝑠) (orange curve) and

𝑜(𝑠)

(𝑠) (blue

curve).

Figure 16 – MDIBC Bode diagram transfer function of 𝐿(𝑠)

(𝑠) and

𝑜(𝑠)

(𝑠)

Figures 17 and 18 show the normalized Bode diagrams of 𝐺𝑖𝑑(𝑠) and 𝐺𝑣𝑑(𝑠),

i.e., 𝐺𝑖𝑑(𝑠)

|𝐺𝑑𝑖| and

𝐺𝑣𝑑(𝑠)

|𝐺𝑑𝑣|. In normalized plot functions, it is easier to identify the shape and

properties of the system response.

Figure 17 – MDIBC Bode diagram of normalized transfer function 𝐿(𝑠)

(𝑠)

47

Figure 18 – MDIBC Bode diagram of normalized transfer function 𝑜(𝑠)

(𝑠)

If the load resistance changes (𝑅), it can affect the transfer function response

due to the position of current inductor zero and the poles are dependent on it. Figure

19 shows how the absolute value of zeros and poles changes when the load varies

from 1 Ω to 1 𝑀Ω. Moreover, it is possible to notice that both parameters stop to

increase when the load resistance reaches the value of 100 Ω.

Figure 19 – Absolute value of zeros and poles as function of the load resistance (𝑅)

48

It is important to point out that, according to (15), the position of the output

voltage zero (𝜔𝑧𝑣) does not depend on the load resistance. The poles, when 1 Ω <

R < 1 𝑀Ω, are pairs of complex conjugate values, figure 20 shows the position of poles

and zeros in the complex plane and, as higher is the load, their position is closer to the

vertical axis (𝑠 = 0).

Figure 20 – Position of zeros and poles as function of the load resistance in the

complex plane.

3.4 Current ripple in fuel cells and batteries

An important converter’s property is, using the same switching frequency (𝑓𝑠),

to increase the frequency of inductor ripple current. A higher frequency of ripple current

inductor has mainly two effects. The first effect is the size and weight reduction of the

power inductor, as discussed previously. The other effect is related with the battery life

span and efficiency [8].

According to the Schneider white paper [19], even if the effects of AC ripple

voltage and current in the batteries are distinct, in excessive amounts, they represent

a potential damage for the battery. The technical bulletin of C&D Technologies Inc., a

battery manufacture [20], relates the AC ripple voltage and current with the

49

temperature rise. If the operating temperature is greater than the designed one,

irreversible damage can occur. The total heat dissipated would be, then, generated

from two different sources, not only the DC current, but also an AC ripple current

resulting in an additional heating effect. Moreover, according to them, the higher the

AC frequency ripple, the lower the impact in the battery.

Regarding the fuel cells, there are disagreements about the effects of current

and voltage ripples. In [21], the authors conclude that the influence of current ripple in

the inverter input (or fuel cell output) is insignificant when the efficiency is concerned.

In [22], the authors could not find any evidence that current ripple – in the frequency

range used – accelerate the deterioration of the fuel cells. In [23], differently, the

authors could conclude that the reduction of the output power is related with the ripple

current content, if the low frequency ripple content in the fuel cell is limited to 15% to

20%, the losses are lower than 1%.

50

4. OTHER CONVERTERS

In this chapter, two other converters topology are presented: the two-phase

interleaved boost converter (IBC) and the multidevice boost converter (MDBC).

Moreover, for these new converters, it is presented their electric circuit, the working

principle when the gate signals are interleaved, the small signal model deduction and

analysis. Their transfer function response curves are also shown.

The proposed converter – multidevice interleaved boost converter (MDIBC) – can

be seen as the exploitation of the properties of both converters presented in this

chapter – two phases and two parallel legs per phase. Reminding that 𝑛 is the number

of phases and 𝑚 is the number of parallel legs per phase, it is possible to identify the

converters studied in this work through the value of these numbers. Other converters

types, not analysed in this work, can be also identified by changing the values of 𝑛 and

𝑚.

Table 8 – Converter type identification through the value of number of phases (𝑛) and

the number of parallel legs per phase (𝑚)

Converter Phases (𝒏) Parallel legs per phase (𝒎)

MDIBC 2 2

IBC 2 1

MDBC 1 2

In this chapter, it is also assumed that the converters operate in the continuous

conduction mode (CCM), the switches have identical duty cycles and the duty cycle

interval is 0 < d < 0.5.

4.1 Two-phase interleaved boost converter (IBC)

In this converter topology, there are two different phases and only one leg of

diode in series with an IGBT per phase. In the output, parallel to the load, there is a

shunt capacitor. The electric circuit of the two – phase interleaved boost converter is

shown in the figure 21.

51

Figure 21 – Electric circuit of the Two-Phase of Interleaved Boost Converter (IBC)

4.1.1 Switching power converter

Figure 22 – IBC Gate sequence for the controllable switches when the duty cycle (𝑑)

is 0.40

52

Table 9 – Description of the IBC conduction of the IGBTs and diodes and load situation

the duty cycle (𝑑) is 0.40 during one period


0 < 𝑡 𝑇𝑠⁄ < 0.40 𝐺1 is conducting 𝐷3 is conducting Powered

0.40 < 𝑡𝑇𝑠⁄ < 0.50 None 𝐷1, 𝐷3 are conducting Powered

0.50 < 𝑡𝑇𝑠⁄ < 0.90 𝐺3 is conducting 𝐷1 is conducting Powered

0.90 < 𝑡𝑇𝑠⁄ < 1 None 𝐷1, 𝐷3 are conducting Powered

4.1.2 Small signal model

The small signal model procedure is derived similarly to the one presented in

the previous chapter. Considering the electric circuit of the figure 21 and the gate signal

pattern from the figure 22, the instants that the drive gate signals G1 and G2 become

zero – t1, t2 respectively, a zero-capacitance resistance – the output voltage (𝑣𝑜) is

equal to the voltage capacitor (𝑣𝑐), it is possible to deduce the small signal model

equations.

As shown in the table 9, during one period, there are four possible states that

the circuit can assume. There are three energy storage elements in the circuit (two

inductors and one capacitor), then three equations are necessary to describe the

circuit: one for each current inductor and one for the voltage capacitor.

The system of equations is described by means of the Kirchhoff voltage law

(KVL) and Kirchhoff current law (KCL). Then, the system of equations is written in

matrix form and the derivative of inductor current and voltage capacitor are isolated in

the left side, i.e., in the same form as: 𝑝𝑥 = 𝐴𝑥 + 𝐵𝑉𝑑, where 𝑝 is the derivative

operator, 𝐴 is a three by three matrix, 𝐵 is a three by one matrix.

53

• Case 1, 0 < t < t1, 𝑟𝑐 ≈ 0, 𝐺1 is on:

𝑉𝑑 − 𝐿. 𝑝𝑖𝐿1 − 𝑟𝐿 . 𝑖𝐿1 = 0𝑉𝑑 − 𝐿. 𝑝𝑖𝐿2 − 𝑟𝐿 . 𝑖𝐿2 − 𝑣𝑜 = 0

𝑖𝐿2 = 𝐶𝑝𝑣𝑜 +𝑣𝑜𝑅

(18)



) =

(

−𝑟𝐿𝐿

0 0

0 −𝑟𝐿𝐿

−1

𝐿

01

𝐶−1

𝑅𝐶)


) +

(

1

𝐿1

𝐿0)

𝑉𝑑

• Case 2, 𝑡1 < t < T𝑠

2, 𝑟𝑐 ≈ 0:



(19)



) =

(

−𝑟𝐿𝐿

0 −1

𝐿

0 −𝑟𝐿𝐿

−1

𝐿1

𝐶

1

𝐶−1

𝑅𝐶)


) +

(

1

𝐿1

𝐿0)

𝑉𝑑

• Case 3, T𝑠

2< t < 𝑡2, 𝑟𝑐 ≈ 0, 𝐺3 is on

𝑉𝑑 − 𝐿. 𝑝𝑖𝐿1 − 𝑟𝐿 . 𝑖𝐿1 − 𝑣𝑜 = 0𝑉𝑑 − 𝐿. 𝑝𝑖𝐿2 − 𝑟𝐿 . 𝑖𝐿2 = 0

𝑖𝐿1 = 𝐶𝑝𝑣𝑜 +𝑣𝑜𝑅

(20)



) =

(

−𝑟𝐿𝐿

0 −1

𝐿

0 −𝑟𝐿𝐿

0

1

𝐶0 −

1

𝑅𝐶)


) +

(

1

𝐿1

𝐿0)

𝑉𝑑

54

• Case 4, 𝑡2 < t < T𝑠, 𝑟𝑐 ≈ 0:



(21)



) =

(

−𝑟𝐿𝐿

0 −1

𝐿

0 −𝑟𝐿𝐿

−1

𝐿1

𝐶

1

𝐶−1

𝑅𝐶)


) +

(

1

𝐿1

𝐿0)

𝑉𝑑

Then, the average value of state variables derivative (p < x >) must be

determined. An index will be written in order to identify which case the matrices are

from: for example, matrices 𝐴 and 𝐵 of the first case will receive an index 1.

p < x >=1

T𝑠∫ 𝑝𝑥. 𝑑𝑡T𝑠

0

(22)

Simplifying the notation, i.e., the 𝑥 = < 𝑥 >, and substituting the values of 𝑝𝑥:

px =1

T𝑠(∫ 𝑝𝑥1. 𝑑𝑡

𝑡1

0


T𝑠2

𝑡1

+∫ 𝑝𝑥3. 𝑑𝑡𝑡2

T𝑠2

+∫ 𝑝𝑥4. 𝑑𝑡T𝑠

𝑡2

)

px =1

T𝑠(∫ (𝐴1𝑥 + 𝐵1𝑣𝑑). 𝑑𝑡

𝑑T𝑠

0

+∫ (𝐴2𝑥 + 𝐵2𝑣𝑑). 𝑑𝑡

T𝑠2

𝑑T𝑠

+∫ (𝐴3𝑥 + 𝐵3𝑣𝑑). 𝑑𝑡T𝑠(

12+𝑑)

T𝑠2

+∫ (𝐴4𝑥 + 𝐵4𝑣𝑑). 𝑑𝑡T𝑠

T𝑠(12+𝑑)

)

px =1

T𝑠((𝐴1𝑥 + 𝐵1𝑣𝑑). 𝑑T𝑠 + (𝐴2𝑥 + 𝐵2𝑣𝑑). (

1

2− 𝑑)T𝑠 + (𝐴3𝑥 + 𝐵3𝑣𝑑). 𝑑T𝑠

+ (𝐴4𝑥 + 𝐵4𝑣𝑑). (1

2− 𝑑) T𝑠)

55

px = (((𝐴1 + 𝐴3)𝑑 +1

2(𝐴2 + 𝐴4)(1 − 2𝑑)) 𝑥

+ ((𝐵1 + 𝐵3)𝑑 +1

2(𝐵2 + 𝐵4)(1 − 2𝑑)) 𝑣𝑑)

(23)

By equation analysis, it is possible to notice that the value “2” in the last equation

is due to the number of phases in the circuit, then it can be substituted by “n”.

px = (((𝐴1 + 𝐴3)𝑑 +1

𝑛(𝐴2 + 𝐴4)(1 − 𝑛𝑑))𝑥

+ ((𝐵1 + 𝐵3)𝑑 +1

𝑛(𝐵2 + 𝐵4)(1 − 𝑛𝑑)) 𝑣𝑑)

(24)

Introduce the separation of AC perturbation and DC component in the variables

𝑥, 𝑑. The letter tilde on top represents the AC perturbation and the capital letter the DC

component. For the variable input voltage is considered pure DC.

𝑥 = + 𝑋𝑑 = + 𝐷

(25)

Substituting (25) into (24), after algebraic manipulations and neglecting higher

harmonics (e.g.: . ≈ 0):

𝑝 = [((𝐴1 + 𝐴3)𝐷 +1

𝑛(𝐴2 + 𝐴4)(1 − 𝑛𝐷))𝑋 + ((𝐵1 + 𝐵3)𝐷 +

1

𝑛(𝐵2 +

𝐵4)(1 − 𝑛𝐷))𝑉𝑑] + [((𝐴1 + 𝐴3)𝐷 +1

𝑛(𝐴2 + 𝐴4)(1 − 𝑛𝐷)) + ((𝐴1 + 𝐴3 −

𝐴2 − 𝐴4)𝑋 + (𝐵1 + 𝐵3 − 𝐵2 − 𝐵4)𝑉𝑑)]

(26)

The first bracket in (26) is the steady state response of the circuit, which, by

definition, is zero value. The matrices 𝐵1, 𝐵2, 𝐵3 and 𝐵4 are equal. Matrices 𝐴2 and 𝐴4

are also equal, so the sum of then is two times the value of 𝐴2, however, in this case it

is also possible to substitute the value “2” by “n”. Rewriting the equation (26):

𝑝 = 𝐴 + (𝐴1 + 𝐴3 − 𝑛𝐴2)𝑋

𝐴 = (𝐴1 + 𝐴3)𝐷 + 𝐴2(1 − 𝑛𝐷)

(27)

56


the left side it is possible to find the transfer function of the state variables. The matrix

“I” is the unit matrix and “s” is the Laplace variable.

(𝑠)

(𝑠)= (𝑠𝐼 − 𝐴)−1(𝐴1 + 𝐴3 − 𝑛𝐴2)𝑋

𝐴 = (𝐴1 + 𝐴3)𝐷 + 𝐴2(1 − 𝑛𝐷)

(28)

Substituting the values, it is possible to find the matrices values.

𝐴 =

(

−𝑟𝐿𝐿

0 −1

𝐿(1 − 𝐷)

0 −𝑟𝐿𝐿

−1

𝐿(1 − 𝐷)

1

𝐶(1 − 𝐷)

1

𝐶(1 − 𝐷) −

1

𝑅𝐶 )

𝐴1 + 𝐴3 − 𝑛𝐴2 =

(

0 0

1

𝐿

0 01

𝐿

−1

𝐶−1

𝐶0)

To find the values of matrix 𝑋 = (𝑥1 𝑥2 𝑥3)𝑡, which are the steady state

variables (𝑥1 = 𝐼𝐿1, 𝑥2 = 𝐼𝐿2, 𝑥3 = 𝑉𝑜), similar equations to the proposed converter are

used. As both topologies – IBC and MDIBC – present two-phases, the procedure to

calculate the current inductor steady state value is the same (equation 12). is

calculated considering that the output power (𝑃𝑜) is equal to the input power (𝑃𝑖) and

the current inductor in both phases are the same. The steady state output voltage is

calculated according to the equation 29, non-ideal characteristics are neglected.

𝑉𝑜 = 𝑉𝑑

1 − 𝐷

(29)

Performing some algebraic calculations, it is found the small signal perturbation

in the current inductor and the voltage capacitor.

• Current inductor state equation:

57




)

∆(𝑠)

(30)

𝐺𝑑𝑖 =

𝑥3 + (𝑥1 + 𝑥2)(1 − 𝐷)𝑅

𝑟𝐿 + 2𝑅(1 − 𝐷)2

𝜔𝑧𝑖 =𝑥3 + (𝑥1+𝑥2)(1 − 𝐷)𝑅

𝑅𝐶𝑥3

∆(𝑠) =𝑠2

𝜔02 +

𝑠

𝑄𝜔0+ 1

𝜔0 = √𝑟𝐿 + 2𝑅(1 − 𝐷)2


√𝑅𝐿𝐶(𝑟𝐿 + 2𝑅(1 − 𝐷)2)






)

∆(𝑠)

(31)

𝐺𝑑𝑣 =

𝑅(2(1 − 𝐷)𝑥3 − (𝑥1 + 𝑥2)𝑟𝐿)

𝑟𝐿 + 2𝑅(1 − 𝐷)2

𝜔𝑧𝑣 =2𝑅(1 − 𝐷)𝑥3 − 𝑅𝑟𝐿(𝑥1 + 𝑥2)

𝑅𝐿(𝑥1 + 𝑥2)

Considering the input voltage (𝑉𝑑) is 200V, the output power (𝑃𝑜) is 30 KW, the

duty cycle (D) is 0.25 and the equations 12 and 29, steady state values of 𝐼𝐿 and 𝑉𝑜 are

calculated and described in the table 10.

Table 10 – Steady state value of 𝐼𝐿 and 𝑉𝑜 for IBC

Parameter Value

𝐼𝐿 (A) 75

𝑉𝑜 (V) 267

The component values (𝑟𝐿, 𝐿, 𝐶 and 𝑅) are substituted according to the table 6,

then numerical transfer functions are found. The position of the zeros in (30) and (31)

is given by 𝜔𝑧𝑖 and 𝜔𝑧𝑣. The poles position, which are the roots of the function ∆(𝑠).

58

These parameters are described in the table 11. Moreover, figures 23 and 24 illustrate

the position of zero and poles in the complex plan for 𝐿(𝑠)

(𝑠) and

𝑜(𝑠)

(𝑠) transfer functions.

Table 11 – Position of the zero and poles of the state equations in the IBC

Parameter Value



𝑃1, 𝑃2 −338 ± 3053𝑗 𝑟𝑎𝑑 𝑠⁄

|𝑃1 | = |𝑃2 | 3071 𝑟𝑎𝑑 𝑠⁄

𝜁 0.1101

Note: The subscript description: 𝑍 – zero, 𝑃 – pole, 𝑖 – inductor current, 𝑣 – output voltage, 𝜁 - damping

ratio


(𝑠)

59


(𝑠)

The transfer function response of 𝐿(𝑠)

(𝑠) (blue curve) and

𝑜(𝑠)

(𝑠) (orange curve) is

illustrated with their respective Bode diagrams in figure 25.

Figure 25 – IBC Bode diagram of the inductor current and output voltage

60

Figures 26 and 27 show the Bode diagrams of normalized transfer functions:

𝐺𝑖𝑑(𝑠)

|𝐺𝑑𝑖| and

𝐺𝑣𝑑(𝑠)

|𝐺𝑑𝑣|.

Figure 26 – IBC Bode diagram of inductor current normalized – unit gain

Figure 27 – IBC Bode diagram of output voltage normalized – unit gain

61

4.2 Multidevice boost converter (MDBC)

In this converter topology, there are one phase and two parallel legs of diode in

series with an IGBT per phase. In the output, parallel to the load, there is an ideal

capacitor in series with the capacitance resistance. The electric circuit of the

multidevice boost converter (MDBC) is shown in the figure 28.

Figure 28 – Electric circuit of the Multidevice boost converter (MDBC)

4.2.1 Switching power converter

Figure 29 – MDBC Gate sequence for the controllable switches when the duty cycle

(𝑑) is 0.40

62

Table 12 – Description of the MDBC conduction of the IGBTs and diodes and load

situation when the duty cycle (𝑑) is 0.40 during one period


0 < 𝑡 𝑇𝑠⁄ < 0.40 𝐺1 is conducting None Not Powered

0.40 < 𝑡𝑇𝑠⁄ < 0.50 None 𝐷1, 𝐷2 are conducting Powered

0.50 < 𝑡𝑇𝑠⁄ < 0.90 𝐺2 is conducting None Not Powered

0.90 < 𝑡𝑇𝑠⁄ < 1 None 𝐷1, 𝐷2 are conducting Powered

4.2.2 Small signal model

Considering the electric circuit and the gate signal pattern (figures 28 and 29),

and the following assumptions the small signal model is deduced. The time 𝑡1 and 𝑡2

are, respectively, the time instants that drive gate signals 𝐺1 and 𝐺2 become zero. The

capacitance resistance is zero: this means that the output voltage (𝑣𝑜) is equal to the

voltage capacitor.

According to the table 12, during one period, there are four possible states that

the circuit can assume. However, as the switches are in the same phase and the duty

cycles of both switches are assumed identical, there are only two possible states that

the MDBC can assume. The states started after the time interval 𝑇𝑠

𝑚 are a repetition of

the previous states.

In this case, there are two energy storage elements in the circuit (an inductor

and a capacitor), hence two equations are necessary to describe the circuit: one for

the current inductor and another for the voltage capacitor.

The system of equations, obtained by means of the Kirchhoff voltage law (KVL)

and Kirchhoff current law (KCL), and the system of equations in matrix form are shown

in the following. In matrix form, the system of equations is written according to 𝑝𝑥 =

𝐴𝑥 + 𝐵𝑉𝑑, where 𝑝 is the derivative operator, 𝐴 is a two by two matrix, 𝐵 is a two by

one matrix.

63

• Case 1, 0 < 𝑡 < 𝑡1, 𝑟𝑐 ≈ 0, 𝐺1 is on:

𝑉𝑑 − 𝑟𝐿𝑖𝐿 − 𝐿. 𝑝𝑖𝐿 = 0−𝑅𝐶𝑝𝑣𝑜 − 𝑣𝑜 = 0

(32)


𝑝 (𝑖𝐿𝑣𝑜) = (

−𝑟𝐿𝐿

0

0 −1

𝑅𝐶

)(𝑖𝐿𝑣𝑜) + (

1

𝐿0

)𝑉𝑑

• Case 2, 𝑡1 < 𝑡 <𝑇𝑠

𝑚, 𝑟𝑐 ≈ 0:

𝑉𝑑 − 𝑟𝐿𝑖𝐿 − 𝐿. 𝑝𝑖𝐿 − 𝑣𝑜 = 0

𝑖𝐿 = 𝐶𝑝𝑣𝑜 −𝑣𝑜𝑅

(33) 𝑝𝑥2 = 𝐴2𝑥2 + 𝐵2𝑉𝑑

𝑝 (𝑖𝐿𝑣𝑜) = (

−𝑟𝐿𝐿

−1

𝐿1

𝐶−1

𝑅𝐶

)(𝑖𝐿𝑣𝑜) + (

1

𝐿0

)𝑉𝑑

Next, the state-space averaging procedure is derived. The matrices with index

1 and 2 are related to the first case and the second case respectively. As there are a

repetition after T𝑠

𝑚, the integration interval is 0 < t <

T𝑠

𝑚.

p < x >=1

𝑇𝑠𝑚

∫ 𝑝𝑥. 𝑑𝑡

𝑇𝑠𝑚

0

(34)

Simplifying the notation, i.e., the 𝑥 = < 𝑥 >, and substituting the values of 𝑝𝑥:

px =1

𝑇𝑠𝑚⁄(∫ 𝑝𝑥1. 𝑑𝑡

𝑡1

0


𝑇𝑠𝑚

𝑡1

)

px =𝑚

𝑇𝑠(∫ (𝐴1𝑥 + 𝐵1𝑣𝑑). 𝑑𝑡

𝑑𝑇𝑠

0

+∫ (𝐴2𝑥 + 𝐵2𝑣𝑑). 𝑑𝑡

𝑇𝑠𝑚

𝑑𝑇𝑠

)

px =𝑚

𝑇𝑠((𝐴1𝑥 + 𝐵1𝑣𝑑). 𝑑𝑇𝑠 + (𝐴2𝑥 + 𝐵2𝑣𝑑). (

1

𝑚− 𝑑)𝑇𝑠)

64

px = (𝐴1𝑚𝐷 + 𝐴2(1 − 𝑚𝐷))𝑋 + (𝐵1𝑚𝐷 + 𝐵2(1 − 𝑚𝐷))𝑉𝑑 (35)

Introduce the separation of AC perturbation and DC component in the variables

𝑥, 𝑑. The letter tilde on top represents the AC perturbation and the capital letter the DC

component. For the variable of the input voltage is considered pure DC.

𝑥 = + 𝑋𝑑 = + 𝐷

(36)

Substituting (36) into (35), after algebraic manipulations and neglecting higher

harmonics (e.g.: . ≈ 0):

𝑝 = [𝐴𝑋 + 𝐵𝑉𝑑] + [𝐴 + ((𝐴1 − 𝐴2)𝑚𝑋 + (𝐵1 − 𝐵2)𝑚𝑉𝑑)] (37)

The first bracket in (37), is the steady state response of the circuit, which, by

definition, is zero. The matrices 𝐵1, 𝐵2 are equal. Considering these prepositions, the

equation 19 can be rewritten.

𝑝 = 𝐴 + 𝑚(𝐴1 − 𝐴2)𝑋

𝐴 = (𝐴1 − 𝐴2)𝑚𝐷 + 𝐴2

(38)


the left side it is possible to find the transfer function of the state variable. The matrix

“𝐼” is the unit matrix and “𝑠” is the Laplace variable.

(𝑠)

(𝑠)= 𝑚(𝑠𝐼 − 𝐴)−1(𝐴1 − 𝐴2)𝑋

𝐴 = (𝐴1 − 𝐴2)𝑚𝐷 + 𝐴2

(38)

Calculating the matrices 𝐴 and (𝐴1 – 𝐴2):

𝐴 = (−𝑟𝐿𝐿

−1

𝐿(1 − 2𝐷)

1

𝐶(1 − 2𝐷) −

1

𝑅𝐶

)

𝐴1 – 𝐴2 = (0

1

𝐿

−1

𝐶0

)

65

To find the values of matrix 𝑋 = (𝑥1 𝑥2)𝑡, which are the steady state variables

(𝑥1 = 𝐼𝐿, 𝑥2 = 𝑉𝑜), similar equations to the proposed converter are used. To calculate

the current inductor steady state value the input (𝑃𝑖) and the output power (𝑃𝑜) are

considered equal. As both topologies – MDBC and MDIBC – present two parallel legs

per phase, and neglecting non-ideal characteristics, the procedure to calculate output

voltage is the same, i.e., equation 13.

𝑃𝑖 = 𝑃𝑜𝑃𝑖 = 𝑉𝑑𝐼𝐿

𝐼𝐿 = 𝑃𝑜𝑉𝑑

(39)

Substituting the matrices 𝐴 and (𝐴1 – 𝐴2) in (38) and performing some algebraic

calculations, it is found the small signal perturbation for both variables.

• Current inductor state equation




)

∆(𝑠)

(40)

𝐺𝑑𝑖 = 2

𝑥2 + 𝑥1(1 − 2𝐷)𝑅

𝑟𝐿 + 𝑅(1 − 4𝐷(1 − 𝐷))

𝜔𝑧𝑖 =𝑥2 + 𝑥1(1 − 2𝐷)𝑅

𝑅𝐶𝑥2

∆(𝑠) =𝑠2

𝜔02 +

𝑠

𝑄𝜔0+ 1

𝜔0 = √𝑟𝐿 + 𝑅(1 − 4𝐷(1 − 𝐷))


√𝑅𝐿𝐶(1 − 4𝐷(1 − 𝐷))






)

∆(𝑠)

(41)

66

𝐺𝑑𝑣 =

2𝑅((1 − 2𝐷)𝑥2 − 𝑥1𝑟𝐿)

𝑟𝐿 + 𝑅(1 − 4𝐷(1 − 𝐷))

𝜔𝑧𝑣 =2𝑅((1 − 2𝐷)𝑥2 − 𝑟𝐿𝑥1)

𝑥1𝐿

Considering the input voltage (𝑉𝑑) is 200V, the output power (𝑃𝑜) is 30 KW, the

duty cycle (D) is 0.25 and the equations 13 and 39, steady state values of 𝐼𝐿 and 𝑉𝑜 are

calculated and described in the table 10.

Table 13 – Steady state value of 𝐼𝐿 and 𝑉𝑜 for MDBC

Parameter Value

𝐼𝐿 (A) 150

𝑉𝑜 (V) 400

The component values (𝑟𝐿, 𝐿, 𝐶 and 𝑅) are substituted according to the table 6

in (40) and (41). The position of the zeros and poles are described in table 14. Figures

30 and 31 the it is illustrated the position of zero and poles in the complex plan.

Table 14 – Position of the zero and poles of the state equations in the MDBC

Parameter Value



𝑃1, 𝑃2 −338 ± 1422𝑗 𝑟𝑎𝑑 𝑠⁄

|𝑃1 | = |𝑃2 | 1461 𝑟𝑎𝑑 𝑠⁄

𝜁 0.2313

Note: The subscript description: 𝑍 – zero, 𝑃 – pole, 𝑖 – inductor current, 𝑣 – output voltage, 𝜁 - damping

ratio

67


(𝑠) for the MDBC


(𝑠) for the MDBC

68

Figure 32 shows the Bode diagram of the two transfer functions: the blue curve

represents the 𝐿(𝑠)

(𝑠) and the orange curve the

𝑜(𝑠)

(𝑠) of MDBC.

Figure 32 – MDBC Bode diagram of the inductor current and output voltage

In the figures 33 and 34, it is shown the normalized Bode diagrams 𝐺𝑖𝑑(𝑠)

|𝐺𝑑𝑖| and

𝐺𝑣𝑑(𝑠)

|𝐺𝑑𝑣| .

69

Figure 33 – MDBC Bode diagram of inductor current normalized – unit gain

Figure 34 – MDBC Bode diagram of output voltage normalized – unit gain

70

5. LOSSES

The evaluation of the losses is important to estimate the overall efficiency of the

converter (42). In this work, it is considered that in the proposed converter there are

three sources of losses: the switching losses (𝑃𝑠𝑤), the diode losses (𝑃𝐷) and the

passive component losses (𝑃𝑐). Their respective equations (43), (44) and (45) are

taken from [8]. Moreover, to perform the loss calculation, manufacture’s datasheets

are need.

𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 (𝜂) =𝑃𝑜𝑢𝑡

𝑃𝑜𝑢𝑡 + 𝑃𝑙𝑜𝑠𝑠𝑒𝑠

(42)

𝑃𝑠𝑤 = (𝐼𝑆𝑟𝑚𝑠2 𝑟𝐶𝐸 + 𝑉𝐶𝐸𝐼𝑠𝑎𝑣 +

𝑉𝑜𝑉𝐶𝐶𝐼𝐶

(𝐸𝑜𝑓𝑓 (𝐼𝑆𝑎𝑣 +∆𝐼

2) + 𝐸𝑜𝑛 (𝐼 −

∆𝐼

2)))

(43)

𝑃𝐷 = 𝐼𝐷𝑟𝑚𝑠2 𝑟𝑓 + 𝑉𝐹0𝐼𝐷𝑎𝑣 +

𝑉𝑜𝑉𝐶𝐶𝐼𝐶

[𝐸𝑟𝑟(𝐼𝐷𝑎𝑣)]𝐹𝑠 (44)

𝑃𝐶 = 𝐼𝐿𝑟𝑚𝑠2 𝑟𝐿 (45)

For the three converters studied, all the components are considered equal and

then they are able to be compared. The value of the losses and their efficiency are

shown in the table 15. It is important to point out that the output voltage is equal to

30 𝑘𝑊. Moreover, it is assumed that only one switch is conducting, and for the MDIBC

there are two diodes conducting at the same time.

Table 15 – Efficiency and converter losses

Converter Switching

losses (𝑃𝑠𝑤)

Diode

losses (𝑃𝐷)

Passive component

losses (𝑃𝐶)

Efficiency

(𝜂)

MDIBC 167.8 𝑊 271.7 𝑊 382.5 𝑊 0.97

IBC 167.8 𝑊 135.8 𝑊 382.5 𝑊 0.98

MDBC 371.2 𝑊 360.8 𝑊 765 𝑊 0.95

71

6. SWITCHING POWER CONTROL

The control strategy used in this work is the dual-loop control, whose control

diagram can be seen in the figure 35. The dual loop control is preferred when

compared to the single loop because it improves stability and increases the system

pass band. The inner loop controls the current inductor and the outer loop the output

voltage.

Figure 35 – Dual loop control strategy adopted.

The control diagram can be split in two different parts, the circuit to be controlled

and the controller. The controlled circuit is represented by the block diagrams 𝐺𝑣𝑑 and

𝐺𝑖𝑑 – output voltage and current inductor state variables – and the block diagram that

considers the switching frequency. The controller is made by the regulators (𝑟𝑒𝑔𝑣 and

𝑟𝑒𝑔𝑖) which are PI controllers, their characteristic equation is shown in (46).

𝑟𝑒𝑔𝑣 𝑜𝑟 𝑖(𝑠) = 𝑘𝑝 +𝑘𝐼𝑠

(46)

The controller design is divided in two parts: firstly, the inner loop is studied –

figure 36 – and the 𝑟𝑒𝑔𝑖 parameters are calculated, then the outer loop and the 𝑟𝑒𝑔𝑣

are also derived.

The procedure and the equations derived in this chapter, shown in the following,

will be determined for a general case that uses a dual loop control strategy. Hence, it

is valid for the three converters studied in this work, i.e., MDIBC, IBC and MDBC. In

the end of the chapter there are the diagrams that illustrate and justify the parameters

choice.

72

Figure 36 – Inner loop control – current inductor control

6.1 Inner loop control

The parameters of 𝑟𝑒𝑔𝑖 is determined through the open loop current transfer

function (𝐿𝐼(𝑠)), which is the product of the three blocks. The mathematical

representation is shown in the (47).

𝐿𝐼(𝑠) = (𝑘𝑝𝐼 +𝑘𝐼𝐼𝑠) (

1

1 + 𝑠𝑇𝑠)𝐺𝑖𝑑

(47)

The equations that represent the circuit to be controlled are merged in one

product 𝐺𝐼 (48) and its Bode diagram is plotted. Then, parameters are identified such

as: the position of zeros, poles and the theoretical cut-off frequency, i.e., frequency

value when the transfer function response LI(s) crosses the 0dB axis.

𝐺𝐼(𝑠) = (1

1 + 𝑠𝑇𝑠)𝐺𝑖𝑑

(48)

The cut-off frequency, theoretically, can assume any value, however, in order to

have a good response and a stable control some design practice must be considered.

The cut-off frequency (𝜔𝑐𝐼) must be at least one decade before the switching

frequency. It is interesting that 𝜔𝑐𝐼 is not positioned in a band that the gain is

−40 𝑑𝐵/𝑑𝑒𝑐, due to the transfer function phase converges quickly to −180 𝑑𝑒𝑔 which

can imply in a positive feedback, hence a non-stable function. Finally, a phase-margin

(𝜑𝑚) must be at least 45 𝑑𝑒𝑔, i.e., the minimum phase distance from −180 𝑑𝑒𝑔 at the

cut-off frequency 𝜔𝑐, must be 45 𝑑𝑒𝑔.

Therefore, with the choice of 𝜔𝑐𝐼 and 𝜑𝑚𝐼, it is possible to derive the 𝑟𝑒𝑔𝑖

parameters. It is imposed that, at 𝜔𝑐𝐼, the open loop transfer function magnitude will

be 1 (or 0 𝑑𝐵) and the phase will be −𝜋 + 𝜑𝑚𝐼, as can be seen in the (49).

73

|𝐿(𝑗𝜔𝑐𝐼)| = 1

∡𝐿(𝑗𝜔𝑐𝐼) = −𝜋 + 𝜑𝑚𝐼

(49)

Substituting (47) in (49) and isolating the parameters 𝑘𝑝𝐼 and 𝑘𝑖𝐼 in the left side,

we have:

|(𝑘𝑝𝐼 +

𝑘𝐼𝐼𝑗𝜔𝑐𝐼

)| |𝐺𝐼(𝑗𝜔𝑐𝐼)| = 1

∡(𝑘𝑝𝐼 +𝑘𝐼𝐼𝑗𝜔𝑐𝐼

) + ∡𝐺𝐼(𝑗𝜔𝑐𝐼) = −𝜋 + 𝜑𝑚𝐼

|(𝑘𝑝𝐼 − 𝑗𝑘𝐼𝐼𝜔𝑐𝐼

)| =1

|𝐺𝐼(𝑗𝜔𝑐𝐼)|

∡ (𝑘𝑝𝐼 − 𝑗𝑘𝐼𝐼𝜔𝑐𝐼

) = −𝜋 + 𝜑𝑚𝐼 − ∡𝐺𝐼(𝑗𝜔𝑐𝐼)

(50)

The vector in (50) can be interpreted in two different ways, using the Cartesian

from, i.e., 𝑧 = 𝑎 + 𝑗𝑏, and in the polar form, i.e., 𝑧 = |𝑧| cos 𝜑 + 𝑗|𝑧| sin𝜑. Comparing

the (50) with the Cartesian form and the polar form, the values of 𝑎, 𝑏, |𝑧| and 𝜑 are

found. Finally, comparing the cartesian and the polar equations, it is possible to

determine the values of 𝑘𝑝𝐼 and 𝑘𝑖𝐼, as shown in the equation 51.

𝑎 = 𝑘𝑝𝐼

𝑏 = −𝑘𝐼𝐼𝜔𝑐𝐼

|𝑧| =1

|𝐺𝐼(𝑗𝜔𝑐𝐼)|

𝜑 = −𝜋 + 𝜑𝑚 − ∡𝐺𝐼(𝑗𝜔𝑐𝐼)

𝑘𝑝𝐼 =1

|𝐺𝐼(𝑗𝜔𝑐𝐼)|cos(−𝜋 + 𝜑𝑚𝐼 − ∡𝐺𝐼(𝑗𝜔𝑐𝐼))

𝑘𝑖𝐼 = −𝜔𝑐

|𝐺𝐼(𝑗𝜔𝑐𝐼)|sin(−𝜋 + 𝜑𝑚𝐼 − ∡𝐺𝐼(𝑗𝜔𝑐𝐼))

(51)

As the parameters of 𝑟𝑒𝑔𝑖 have already been calculated, it is necessary to

calculate the closed loop transfer function. In the figure 37 (a), it is shown the

simplification of the inner loop block diagram (figure 36). If two blocks diagram is used,

it is easier to derive the closed loop current transfer function (𝐾𝐼(𝑠)) using this strategy.

The figure 37 (b) shows another block simplification, with the interpretation of the inner

closed loop when the outer loop is analysed.

74

(a) (b)

Figure 37 – Simplification of the inner loop control

From the figure 37 (a), it is possible to find a relation between the output (𝑑) and

the input (𝑖𝐿 𝑟𝑒𝑓), as illustrated in the (52).

(𝑠) = 𝐴(𝑠)(𝑖𝐿 𝑟𝑒𝑓(𝑠) − (𝑠)𝐵(𝑠))

(𝑠) =

𝐴(𝑠)

1 + 𝐴(𝑠)𝐵(𝑠)𝑖𝐿 𝑟𝑒𝑓(𝑠)

𝐾𝐼(𝑠) =𝐴(𝑠)

1 + 𝐴(𝑠)𝐵(𝑠)

(52)

If the same notation of the figure 37 (a) is used, the open loop current transfer

function (𝐿𝐼(𝑠)) is the product of the two blocks 𝐴𝐵. Hence, it is possible to relate the

closed loop transfer function (𝐾𝐼(𝑠)) and the open loop transfer function (𝐿𝐼(𝑠)), as

shown in the (53).

𝐾𝐼(𝑠) =𝐴(𝑠)

1 + 𝐴(𝑠)𝐵(𝑠)

𝐾𝐼(𝑠) =

𝐴(𝑠)𝐵(𝑠)𝐵(𝑠)

1 + 𝐴(𝑠)𝐵(𝑠)

𝐾𝐼(𝑠) =

𝐿𝐼(𝑠)𝐺𝑖𝑑(𝑠)

1 + 𝐿𝐼(𝑠)

(53)

6.2 Outer loop control

The procedure to calculate the parameters of 𝑟𝑒𝑔𝑣 is similar to the calculation

for the inner loop control. As previously, it is necessary to calculate the open loop

voltage transfer function (𝐿𝑉(𝑠)). The inner loop is represented by the KI block diagram

75

– figure 37 (b), the value of the 𝐾𝐼(𝑠) is shown in the (53). The equation (54) shows

the open loop voltage transfer function.

𝐿𝑉(𝑠) = (𝑘𝑝𝑉 +𝑘𝐼𝑉𝑠)𝐾𝑉(𝑠)𝐺𝑣𝑑

(54)

The bode diagram of (55) is plotted, and the position of poles and zeros are

analysed in order to choose a cut-off frequency (𝜔𝑐𝑉).

𝐺𝑉(𝑠) = 𝐾𝐼(𝑠)𝐺𝑣𝑑 (55)

However, besides the design practices pointed out for the design of the inner

loop control, the voltage cut-off frequency was chosen according to the (56).

𝜔𝑐𝑉 =𝜔𝑐𝐼10

(56)

If the parameters 𝜔𝑐𝑉 and 𝜑𝑚𝑉 are chosen, it is possible to derive the 𝑟𝑒𝑔𝑣

parameters using the same step-by-step from the inner loop – equations (49) to (51).

Finally, it is necessary to calculate the closed loop voltage transfer function; as

it is a simple feedback loop it is easy to determine, as can be shown in the (57).

𝐾𝑉(𝑠) =𝐿𝑉(𝑠)

1 + 𝐿𝑉(𝑠)

(57)

The values of the voltage and current cut-off frequency for the three converter

topologies is shown in the table 16.

Table 16 – Values of current and voltage cut-off frequency for each converter type.

Converter Current cut-off frequency (𝝎𝒄𝑰) Voltage cut-off frequency (𝝎𝒄𝑽)

MDIBC 6280 𝑟𝑎𝑑/𝑠 628 𝑟𝑎𝑑/𝑠

IBC 6280 𝑟𝑎𝑑/𝑠 628 𝑟𝑎𝑑/𝑠

MDBC 3140 𝑟𝑎𝑑/𝑠 314 𝑟𝑎𝑑/𝑠

The value of the phase-margin (𝜑𝑚), for a given 𝜔𝑐, changes the value of 𝑘𝑝

and 𝑘𝐼; therefore, to avoid negative values of 𝑘𝑝 and 𝑘𝐼 the graphs 𝑘𝑝 𝑣𝑠. 𝜑𝑚 and

𝑘𝑖 𝑣𝑠. 𝜑𝑚 are plotted. The tables 14, 15 and 16 show the values of the controller

76

proportional and integral parameters (𝑘𝑝𝐼 , 𝑘𝑖𝐼 𝑘𝑝𝑉, 𝑘𝑖𝑉) for, respectively, the MDIBC,

IBC and MDBC.


(𝑘𝑝𝐼 , 𝑘𝑖𝐼 𝑘𝑝𝑉, 𝑘𝑖𝑉) for the MDIBC when 𝜑𝑚𝐼 = 60 𝑑𝑒𝑔, 𝜑𝑚𝑉 = 70 𝑑𝑒𝑔

Parameter Value

𝑘𝑝𝐼 0.0027

𝑘𝑖𝐼 2.64

𝑘𝑝𝑉 0. 384

𝑘𝑖𝑉 626

Figure 38 – 𝑘𝑝𝐼 𝑣𝑠. 𝜑𝑚𝐼 and 𝑘𝑖𝐼 𝑣𝑠. 𝜑𝑚𝐼 – MDIBC

77

Figure 39 – 𝑘𝑝𝑉 𝑣𝑠. 𝜑𝑚𝑉 and 𝑘𝑖𝑉 𝑣𝑠. 𝜑𝑚𝑉 – MDIBC


(𝑘𝑝𝐼 , 𝑘𝑖𝐼 𝑘𝑝𝑉, 𝑘𝑖𝑉) for the IBC when 𝜑𝑚𝐼 = 60 𝑑𝑒𝑔, 𝜑𝑚𝑉 = 60 𝑑𝑒𝑔

Parameter Value

𝑘𝑝𝐼 0.0068

𝑘𝑖𝐼 2.81

𝑘𝑝𝑉 0.0257

𝑘𝑖𝑉 475

78

Figure 40 – 𝑘𝑝𝐼 𝑣𝑠. 𝜑𝑚𝐼 and 𝑘𝑖𝐼 𝑣𝑠. 𝜑𝑚𝐼 – IBC

Figure 41 – 𝑘𝑝𝑉 𝑣𝑠. 𝜑𝑚𝑉 and 𝑘𝑖𝑉 𝑣𝑠. 𝜑𝑚𝑉 – IBC

79


(𝑘𝑝𝐼 , 𝑘𝑖𝐼 𝑘𝑝𝑉, 𝑘𝑖𝑉) for the MDBC when 𝜑𝑚𝐼 = 70 𝑑𝑒𝑔, 𝜑𝑚𝑉 = 70 𝑑𝑒𝑔

Parameter Value

𝑘𝑝𝐼 0.0011

𝑘𝑖𝐼 0.372

𝑘𝑝𝑉 0.175

𝑘𝑖𝑉 250

Figure 42 – 𝑘𝑝𝐼 𝑣𝑠. 𝜑𝑚𝐼 and 𝑘𝑖𝐼 𝑣𝑠. 𝜑𝑚𝐼 – MDBC

80

Figure 43 – 𝑘𝑝𝑉 𝑣𝑠. 𝜑𝑚𝑉 and 𝑘𝑖𝑉 𝑣𝑠. 𝜑𝑚𝑉 – MDBC

The Bode diagram of the transfer functions: 𝐺𝐼(𝑠), 𝐿𝐼(𝑠), 𝐾𝐼(𝑠), 𝐺𝑉(𝑠), 𝐿𝑉(𝑠),

𝐾𝑉(𝑠), for each converter topology, are shown in the figures 44 to 55. These figures

illustrate the procedure described in this chapter to find the controller parameters (𝑘𝑝

and 𝑘𝑖) for each controller (𝑟𝑒𝑔𝑖 and 𝑟𝑒𝑔𝑣).

81

• MDIBC

Figure 44 – Transfer function GI – MDIBC

Figure 45 – Open loop LI(s) and closed loop KI(s) transfer functions – MDIBC

82

Figure 46 – Transfer function GV – MDIBC

Figure 47 – Open loop LV(s) and closed loop KV(s) transfer functions – MDIBC

83

• IBC

Figure 48 – Transfer function GI – IBC

Figure 49 – Open loop LI(s) and closed loop KI(s) transfer functions – IBC

84

Figure 50 – Transfer function GV – IBC

Figure 51 – Open loop LV(s) and closed loop KV(s) transfer functions – IBC

85

• MDBC

Figure 52 – Transfer function GI – MDBC

Figure 53 – Open loop LI(s) and closed loop KI(s) transfer functions – MDBC

86

Figure 54 – Transfer function GV – MDBC

Figure 55 – Open loop LV(s) and closed loop KV(s) transfer functions – MDBC

87

7. SIMULATION AND RESULTS

The simulations are performed in Matlab/Simulink. The simulation can be

divided in two parts, the electric circuit and the controller. In the electric circuit two

circuit breakers are included: one between the capacitor and the diode outputs, the

other circuit breaker is between the capacitor and the load. These circuit breakers are

used to charge the capacitor before to be connected to the load, as illustrated in the

electric circuit of the MDIBC (figure 56).

Figure 56 – Electric circuit of the MDIBC in Simulink

To guarantee that the switching will start only after the capacitor is charged, the

enable block starts only after the signal 𝑡_𝑠𝑡𝑎𝑟𝑡, as can be seen in the figure 57.

Moreover, after each output from the regulation block there is a 2-input Logic AND

Gate, in which the second input is the signal 𝑡_𝑠𝑡𝑎𝑟𝑡. The output of the logic gate AND

is the gate signal of the IGBTs; this logic is illustrated in the figure 58. This means that

the drive gate signals will only be different than zero after the time 𝑡_𝑠𝑡𝑎𝑟𝑡.

Figure 57 – Regulator block

88

Figure 58 – Start logic for the drive gate signal

The controller part is shown in the figure 59.

Figure 59 – Controller diagram MDIBC in Simulink

From the figure 59, it is possible to identify properties and the blocks from the

controller part. Firstly, the regulators are not simple PI transfer functions blocks, but

anti-windup PI controllers as shown in the figure 60. This solution was chosen because

it guarantees stability. As a saturation block is added to the controller diagram, the

output is limited, and it avoids the divergence of the integral error, hence the integral

error is kept small. For the voltage controller, in the saturation block the limit value is

set at the steady stated current value (𝐼𝐿). For the current controller, the saturation

block limit is the input voltage (𝑉𝑑).

89

Figure 60 – Anti-windup PI controller

From the output of the voltage regulator, two branches are derived, one for each

current inductor (𝑖𝐿1 and 𝑖𝐿2). The output of the current regulator is divided by a constant

value, in this case, the input voltage 𝑉𝑑. This ratio is the voltage control and it is limited

to 0 < 𝑣𝑐 < 1. From the output of the ratio block, two control signals are originated, one

for each parallel leg per phase. Finally, these voltages control are compared to the

same saw-tooth wave (𝑣𝑠𝑡), but time delayed in 𝑇𝑠𝑤

4 from each other, i.e., 𝑣𝑐1 is

compared to 𝑣𝑠𝑡, the 𝑣𝑐3 is compared to the 𝑣𝑠𝑡 delayed by 𝑇𝑠𝑤

4, the 𝑣𝑐2 is compared to

𝑣𝑠𝑡 delayed by 𝑇𝑠𝑤

2 and 𝑣𝑐4 is compared to 𝑣𝑠𝑡delayed by

3

4𝑇𝑠𝑤. If 𝑣𝑐 > 𝑣𝑠𝑡, it is applied

a logic signal equal to one to the gate, if 𝑣𝑐 < 𝑣𝑠𝑡, a logic signal equal to zero is applied.

It is important to notice that the index of the 𝑣𝑐 represents which IGBT is being

controlled, for example: 𝑣𝑐1 controls the gate signal of 𝐺1. The comparison of the

control voltage 𝑣𝑐 with 𝑣𝑠𝑡 delayed in 360𝑜

𝑚𝑛 provides the interleaved control strategy.

The other converters, IBC (figures 61 and 62) and MDBC (figures 63 and 64),

have the similar work principle.

In the case of converter which presents two phases, MDIBC and IBC, the output

of the voltage regulator, 𝑖𝐿 𝑟𝑒𝑓, is divided in two branches, one for each phase. For the

converters which presents two parallel legs per phase, MDIBC and MDBC, the control

voltage is the same, but they are compared to different carrier signals.

90

Figure 61 – Electric circuit of the IBC in Simulink


91

Figure 63 – Electric circuit of the IBC in Simulink


To observe the behaviour of the output voltage, the output voltage reference

(𝑣𝑜 𝑟𝑒𝑓) is simulated as ramp source voltage. A saturation block is connected to the

ramp source, upper limit is set as the steady state output voltage. This limit is different

for each converter topology as shown in the tables 5, 10 and 13.

Figure 65 shows the variation of 𝑣𝑜 𝑟𝑒𝑓 (orange curve) and 𝑣𝑜 (blue curve) with

respect to time (𝑡).

Figure 65 – 𝑣𝑜 vs. t (orange curve) and 𝑣𝑜 𝑟𝑒𝑓 vs. t (blue curve) for the MDIBC

It is possible to notice that the 𝑣𝑜 does not reach the 400 V as indicated 𝑣𝑜 𝑟𝑒𝑓.

This can be explained from (13), in which to calculate 𝑣𝑜 non-ideal parameters were

neglected, such as the line resistance, the IGBTs and diodes resistance. Observe that

the orange line reaches a value close to 𝑉𝑑 in 0.025 seconds, which is the time to

92

charge the shunt capacitor. When the time is equal to 0.2 seconds, 𝑣𝑜 starts to increase

and the regulators are enabled. Then, after one second, the circuit breaker between

the shunt capacitor and the load closes, in this time instant, a ripple in the 𝑣𝑜 is

observed. Figure 66 shows 𝑣𝑜 in the steady state condition, from the graph 𝑣𝑜 =

394.72 ± 0.02 [𝑉], hence, the voltage ripple is equal to 0.0051%.

Figure 66 – 𝑣𝑜vs. t (orange curve) and 𝑣𝑜 𝑟𝑒𝑓 vs. t (blue curve) for the MDIBC in steady

state condition for the MDIBC

In the figure 67, it is shown the current from the source (green curve) and the

inductor currents (orange curve). The average value of the source current (𝑖𝑑) is, as

predicted, the double of the current inductor (𝑖𝐿) value. To calculate the ripple of the

current inductor, it is considered the period in which the switch is off and the

resistances are neglected. For the three configurations the equation from KVL is the

same (equation 58). The main difference among the three configurations is the

integration interval.

93

𝑉𝑑 − 𝐿𝑑𝑖𝐿𝑑𝑡

− 𝑉𝑜 = 0 (58)

Then, isolating the variation of the current inductor in the left side and integrating

it with respect to time, it is possible to calculate the current ripple for the three cases.

The equations 59, 60 and 61 show the formulas to calculate the current ripple for the

MDIBC, IBC and MDBC respectively.

∆𝑖𝐿− =

𝑉𝑑 − 𝑉𝑜𝐿

∫ 𝑑𝑡

𝑇𝑠𝑚𝑛

𝑑.𝑇𝑠

=𝑉𝑑 − 𝑉𝑜𝑚𝑛𝐿𝑓𝑠

(1 −𝑚𝑛𝑑) (59)

∆𝑖𝐿− =


∫ 𝑑𝑡

𝑇𝑠𝑛

𝑑.𝑇𝑠

=𝑉𝑑 − 𝑉𝑜𝑛𝐿𝑓𝑠

(1 − 𝑛𝑑) (60)

∆𝑖𝐿− =


∫ 𝑑𝑡

𝑇𝑠𝑚

𝑑.𝑇𝑠

=𝑉𝑑 − 𝑉𝑜𝑚𝐿𝑓𝑠

(1 − 𝑚𝑑) (61)

Moreover, it is possible to notice from the figures that the source current ripple

is lower than the inductor current ripple. This phenomenon can be explained from the

interleaved current between the two phases, as shown in the figure 68. The reduction

of current inductor ripples has a benefit effect in batteries or fuel cells, due to a lower

current ripple implies in a lower heat source, which could damage the source, and,

hence, a higher the life span.

94


MDIBC

Figure 68 – 𝑖𝐿1𝑣𝑠. 𝑡 (blue curve) and 𝑖𝐿2𝑣𝑠. 𝑡 for MDIBC (orange curve) for MDIBC

Figure 69 – 𝑖𝑑 𝑣𝑠. 𝑡 for MDIBC

Comparing the figures 68 and 69, the frequency of the source current is twice

the frequency of the inductor, as predicted in chapter 3. Moreover, from the same

95

figures, 𝑖𝐿1 = 𝑖𝐿2 = 75.0 ± 3.5 [A] and 𝑖𝑑 = 150 ± 0.025 [𝐴]. Therefore, the inductor

current ripple is limited to 4.6% and the source current ripple to 0.02%.

Figure 70 to 73 show the graphs of the output voltage and current inductor to

the IBC.

Figure 70 – 𝑣𝑜vs. t (orange curve) and 𝑣𝑜 𝑟𝑒𝑓 vs. t (blue curve) for the IBC in steady

state condition


IBC

96

Figure 72 – 𝑖𝐿1𝑣𝑠. 𝑡 (blue curve) and 𝑖𝐿2𝑣𝑠. 𝑡 for MDIBC (orange curve) for IBC

Figure 73 – 𝑖𝑑 𝑣𝑠. 𝑡 for IBC

From the figure 70, 𝑣𝑜 = 396 ± 0.053 [𝑉], therefore the output voltage ripple is

limited to 0.013%. According to the figures 72 and 73, the total current and current

inductor are equal to 𝑖𝑑 = 150.0 ± 0.80 [𝐴] and 𝑖𝐿1 = 𝑖𝐿2 = 75 ± 7 [𝐴]. Therefore, the

source current and the inductor current ripple are limited to 0.53% and 9.33%.

97

The characteristics of the MDBC are shown in the figure 74 to 75 show the

graphs of the output voltage and current inductor to the MDBC.

Figure 74 – 𝑣𝑜vs. t (blue curve) and 𝑣𝑜 𝑟𝑒𝑓 vs. t (orange curve) for the MDBC in steady

state condition

Figure 75 – 𝑖𝑑 𝑣𝑠. 𝑡 for MDBC

Figure 74 shows the output voltage when the converter is in steady state,

according to the figure the value of 𝑣𝑜 = 393.0 ± 1.3 [𝑉]. The value of the current

98

inductor, as reported by the figure 75, is 𝑖𝐿 = 150.0 ± 3.3 [A]. The output voltage ripple

is limited to 0.33%, the inductor current ripple is equal to 2.2%.

Table 20 describes the values of the output voltage ripple (𝑣𝑜), the source

current (𝑖𝑑) and the current inductor (𝑖𝐿) ripple for each topology.

Table 20 – Output voltage, total current and inductor current ripple for each topology

Converter

topology

Output voltage

ripple (∆𝑽𝒐

𝑽𝒐)

Source current

(∆𝑰𝒅

𝑰𝒅)

Current inductor

(∆𝑰𝑳

𝑰𝑳)

MDIBC 0.0051% 0.02% 4.6%

IBC 0.013% 0.53% 9.33%

MDBC 0.33% 2.2%. 2.2%

From the table 20, it is possible to notice that the proposed converter ripple

values are the half when compared to the IBC and much lower when compared to the

MDBC. These values of output voltage and current inductor ripple are in accordance

with the assumptions used to calculate the component values.

99

8. CONCLUSION

In this work, it is presented a unidirectional DC/DC converter, the multidevice

interleaved boost converter (MDIBC), which would be a suitable choice to optimize the

drive system fed by fuel cell. This converter is compared to other two converter

topologies, the interleaved boost converter (IBC) and the multidevice boost converter

(MDBC). For the three topologies, it was analysed the switching characteristic and

boundaries. Moreover, the small signal model and losses analysis is deduced. Finally,

it is designed a dual loop control strategy to perform the simulations.

The simulations show that, using the same component values, the MDIBC can

increase the current ripple frequency and reduce the value of the output voltage and

the current ripple is significantly which would imply a size reduction of the passive

components, therefore the size and weight reduction of the converter. A lower current

inductor ripple would also slow the aging process in the fuel cell.

In a nutshell, regarding the efficiency and the performance of the converter, it

seems to be a good solution for automotive application, when compared to the other

two converter topologies.

100

9. REFERENCE

[1] R. Matulka, “The History of the Electric Car”. U.S. Department of Energy.

September 15th, 2014. Available in: https://www.energy.gov/. Access in: November

12th, 2018.

[2] ELECTRONIC DICTIONARY Merriam-Webster Unabridged. Springfield,

Massachusetts, EUA. Available in: http://unabridged.merriam-webster.com/. Access

in: November 12th, 2018.

[3] R. A. Dougal, S. Liu and R. E. White, "Power and life extension of battery-

ultracapacitor hybrids," in IEEE Transactions on Components and Packaging

Technologies, vol. 25, no. 1, pp. 120-131, March 2002.

[4] O. Hegazy, M. El Baghdadi, J. Van Mierlo, P. Lataire and T. Coosemans, "Analysis

and modelling of a bidirectional multiport DC/DC power converter for battery electric

vehicle applications," 2014 16th European Conference on Power Electronics and

Applications, Lappeenranta, 2014, pp. 1-12.

[5] O. Hegazy, R. Barrero, J. Van Mierlo, P. Lataire, N. Omar and T. Coosemans, "An

Advanced Power Electronics Interface for Electric Vehicles Applications," in IEEE

Transactions on Power Electronics, vol. 28, no. 12, pp. 5508-5521, Dec. 2013.

[6] J. W. Shim, Y. Cho, S. Kim, S. W. Min and K. Hur, "Synergistic Control of SMES

and Battery Energy Storage for Enabling Dispatchability of Renewable Energy

Sources," in IEEE Transactions on Applied Superconductivity, vol. 23, no. 3, pp.

5701205-5701205, June 2013, Art no. 5701205.

[7] F. Akar, Y. Tavlasoglu, E. Ugur, B. Vural and I. Aksoy, "A Bidirectional Nonisolated

Multi-Input DC–DC Converter for Hybrid Energy Storage Systems in Electric Vehicles,"

in IEEE Transactions on Vehicular Technology, vol. 65, no. 10, pp. 7944-7955, Oct.

2016.

[8] O. Hegazy, J. V. Mierlo and P. Lataire, "Analysis, Modeling, and Implementation of

a Multidevice Interleaved DC/DC Converter for Fuel Cell Hybrid Electric Vehicles," in

IEEE Transactions on Power Electronics, vol. 27, no. 11, pp. 4445-4458, Nov. 2012.


101

[9] O. Hegazy, J. Van Mierlo and P. Lataire, "Analysis, control and comparison of

DC/DC boost converter topologies for fuel cell hybrid electric vehicle applications,"

Proceedings of the 2011 14th European Conference on Power Electronics and

Applications, Birmingham, 2011, pp. 1-10.

[10] O. Ellabban, J. Van Mierlo, P. Lataire and P. Van den Bossche, "Z-source inverter

for vehicular applications," 2011 IEEE Vehicle Power and Propulsion Conference,

Chicago, IL, 2011, pp. 1-6.

[11] O. Hegazy, J. Van Mierlo and P. Lataire, "Design and control of bidirectional

DC/AC and DC/DC converters for plug-in hybrid electric vehicles," 2011 International

Conference on Power Engineering, Energy and Electrical Drives, Malaga, 2011, pp. 1-

7.

[12] J. W. Zillman, “A History of Climate Activities”. World Meteorological Organization.

July 2009. Available in: https://public.wmo.int/en. Access in: November 15th, 2018.

[13] “The Paris Agreement”. World Meteorological Organization. July 2009. Available

in: https://unfccc.int/. Access in: November 15th, 2018.

[14] “Global Greenhouse Gas Emissions Data”. United States Environmental

Protection Agency. Available in: https://www.epa.gov/. Access in: November 16th,

2018.

[15] “Reducing Pollution with Electric Vehicles”. U.S. Department of Energy. Available

in: https://www.energy.gov/. Access in: November 16th, 2018.

[16] “Alternative Fuels Data Center - Emissions from Hybrid and Plug-In Electric

Vehicles”. U.S. Department of Energy. Available in: https://www.energy.gov/. Access

in: November 16th, 2018.

[17] D. L. Chandler, “Can today’s EVs make a dent in climate change?”. MIT News

Office. August 15, 2016. Available in: http://news.mit.edu/. Access in: November 16th,

2018.

[18] N. Mohan, Tore M. Undeland, William P. Robbins, “Power electronics Converters,

Applications and Design”, 3rd edition. Willey, 2002.



http://news.mit.edu/

102

[19] Dan Lambert , “Lead Acid Battery Lifecycle: Terms and Definitions”. Schneider

Electric – Data Center Science Center.

[20] C&D TECHNOLOGIES, INC., “Charger Output AC Ripple Voltage and the affect

on VRLA batteries”. Technical Bulletin, 2012.

[21] A. Purvins, O. Krievs, I. Steiks and L. Ribickis, "Influence of the current ripple on

the hydrogen fuel cell powered inverter system efficiency," 2009 13th European

Conference on Power Electronics and Applications, Barcelona, 2009, pp. 1-7.

[22] W. Choi, P. N. Enjeti, J. W. Howze, G. Joung ,“An experimental evaluation of the

effects of ripple current generated by the power conditioning stage on a proton

exchange membrane fuel cell stack”, Journal of Materials Engineering and

Performance, 2004, Volume 13, Number 3, Page 257.

[23] L. Schindele, M. Braun and H. Spath, "The influence of power electronic dynamics

on PEM fuel cell-system," 2005 European Conference on Power Electronics and

Applications, Dresden, 2005, pp. 9 pp.-P.9.

POLITECNICO DI MILANO Scuola di Ingegneria Industriale e …...Tanio, Lucas Yugo Converters for...

Documents

Transcript of POLITECNICO DI MILANO Scuola di Ingegneria Industriale e …...Tanio, Lucas Yugo Converters for...